mrsstuckeysmathclass.weebly.commrsstuckeysmathclass.weebly.com/uploads/6/1/1/2/... · © Glencoe/McGraw-Hill iii Glencoe Algebra 2 Contents Vocabulary Builder . . . . . . . .

Chapter 10Resource Masters

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9

ANSWERS FOR WORKBOOKS The answers for Chapter 10 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-828013-3 Algebra 2Chapter 10 Resource Masters

1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02

© Glencoe/McGraw-Hill iii Glencoe Algebra 2

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 10-1Study Guide and Intervention . . . . . . . . 573–574Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 575Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 576Reading to Learn Mathematics . . . . . . . . . . 577Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 578






Chapter 10 AssessmentChapter 10 Test, Form 1 . . . . . . . . . . . 609–610Chapter 10 Test, Form 2A . . . . . . . . . . 611–612Chapter 10 Test, Form 2B . . . . . . . . . . 613–614Chapter 10 Test, Form 2C . . . . . . . . . . 615–616Chapter 10 Test, Form 2D . . . . . . . . . . 617–618Chapter 10 Test, Form 3 . . . . . . . . . . . 619–620Chapter 10 Open-Ended Assessment . . . . . 621Chapter 10 Vocabulary Test/Review . . . . . . 622Chapter 10 Quizzes 1 & 2 . . . . . . . . . . . . . . 623Chapter 10 Quizzes 3 & 4 . . . . . . . . . . . . . . 624Chapter 10 Mid-Chapter Test . . . . . . . . . . . . 625Chapter 10 Cumulative Review . . . . . . . . . . 626Chapter 10 Standardized Test Practice . 627–628Unit 3 Test/Review (Ch. 8–10) . . . . . . . 629–630

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A30

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Teacher’s Guide to Using theChapter 10 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 10 Resource Masters includes the core materialsneeded for Chapter 10. These materials include worksheets, extensions, andassessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 10-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Algebra 2

Assessment OptionsThe assessment masters in the Chapter 10Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 572–573. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

1010

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 10.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

Change of Base Formula

common logarithm

LAW·guh·RIH·thuhm

exponential decay

EHK·spuh·NEHN·chuhl

exponential equation

exponential function

exponential growth

exponential inequality

(continued on the next page)

© Glencoe/McGraw-Hill vii Glencoe Algebra 2

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

Vocabulary Term Found on Page Definition/Description/Example

logarithm

logarithmic function

LAW·guh·RIHTH·mihk

natural base, e

natural base exponential function

natural logarithm

natural logarithmic function

rate of decay

rate of growth

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

1010

Study Guide and InterventionExponential Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1

© Glencoe/McGraw-Hill 573 Glencoe Algebra 2

Less

on

10-

1

Exponential Functions An exponential function has the form y ! abx,where a " 0, b # 0, and b " 1.

1. The function is continuous and one-to-one.

Properties of an2. The domain is the set of all real numbers.

Exponential Function3. The x-axis is the asymptote of the graph.4. The range is the set of all positive numbers if a # 0 and all negative numbers if a $ 0.5. The graph contains the point (0, a).

Exponential Growth If a # 0 and b # 1, the function y ! abx represents exponential growth.and Decay If a # 0 and 0 $ b $ 1, the function y ! abx represents exponential decay.

Sketch the graph of y ! 0.1(4)x. Then state the function’s domain and range.Make a table of values. Connect the points to form a smooth curve.

The domain of the function is all real numbers, while the range is the set of all positive real numbers.

Determine whether each function represents exponential growth or decay.a. y ! 0.5(2)x b. y ! %2.8(2)x c. y ! 1.1(0.5)x

exponential growth, neither, since %2.8, exponential decay, sincesince the base, 2, is the value of a is less the base, 0.5, is betweengreater than 1 than 0. 0 and 1

Sketch the graph of each function. Then state the function’s domain and range.

1. y ! 3(2)x 2. y ! %2! "x3. y ! 0.25(5)x

Domain: all real Domain: all real Domain: all real numbers; Range: all numbers; Range: all numbers; Range: allpositive real numbers negative real numbers positive real numbers

Determine whether each function represents exponential growth or decay.

4. y ! 0.3(1.2)x growth 5. y ! %5! "xneither 6. y ! 3(10)%x decay4

&5

x

y

O

x

y

O

x

y

O

1&4

x %1 0 1 2 3

y 0.025 0.1 0.4 1.6 6.4

x

y

O

Example 1Example 1

Example 2Example 2

ExercisesExercises


Exponential Equations and Inequalities All the properties of rational exponentsthat you know also apply to real exponents. Remember that am ' an ! am ( n, (am)n ! amn,and am ) an ! am % n.

Property of Equality for If b is a positive number other than 1,Exponential Functions then bx ! by if and only if x ! y.

Property of Inequality forIf b # 1

Exponential Functionsthen bx # by if and only if x # yand bx $ by if and only if x $ y.

Study Guide and Intervention (continued)

Exponential Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1

Solve 4x " 1 ! 2x # 5.4x % 1 ! 2x ( 5 Original equation

(22)x % 1 ! 2x ( 5 Rewrite 4 as 22.

2(x % 1) ! x ( 5 Prop. of Inequality for ExponentialFunctions

2x % 2 ! x ( 5 Distributive Property

x ! 7 Subtract x and add 2 to each side.

Solve 52x " 1 $ .

52x % 1 # Original inequality

52x % 1 # 5%3 Rewrite as 5%3.

2x % 1 # %3 Prop. of Inequality for Exponential Functions

2x # %2 Add 1 to each side.

x # %1 Divide each side by 2.

The solution set is {x|x # %1}.

1&125

1&125

1%125

Example 1Example 1 Example 2Example 2

ExercisesExercises

Simplify each expression.

1. (3#2$)#2$ 2. 25#2$ ' 125#2$ 3. (x#2$y3#2$)#2$

9 55!2" or 3125!2" x2y6

4. (x#6$)(x#5$) 5. (x#6$)#5$ 6. (2x*)(5x3*)x!6" # !5" x!30" 10x4&

Solve each equation or inequality. Check your solution.

7. 32x % 1 ! 3x ( 2 3 8. 23x ! 4x ( 2 4 9. 32x % 1 ! "

10. 4x ( 1 ! 82x ( 3 " 11. 8x % 2 ! 12. 252x ! 125x ( 2 6

13. 4#x$ ! 16#5$ 20 14. x#3$ ! 36%&&34& 6 15. x#2$ ! 81&#18$

&3

16. 3x % 4 $ x ' 1 17. 42x % 2 # 2x ( 1 x $ 18. 52x $ 125x % 5 x $ 15

19. 104x ( 1 # 100x % 2 20. 73x $ 49x2 21. 82x % 5 $ 4x ( 8

x $ " x $ or x ' 0 x ' %341%

3%2

5%2

5%3

1&27

2%3

1&16

7%4

1%2

1&9

Skills PracticeExponential Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1


Less

on

10-

1


1. y ! 3(2)x 2. y ! 2! "x

domain: all real numbers; domain: all real numbers;range: all positive numbers range: all positive numbers


3. y ! 3(6)x growth 4. y ! 2! "xdecay

5. y ! 10%x decay 6. y ! 2(2.5)x growth

Write an exponential function whose graph passes through the given points.

7. (0, 1) and (%1, 3) y ! # $x8. (0, 4) and (1, 12) y ! 4(3)x

9. (0, 3) and (%1, 6) y ! 3# $x10. (0, 5) and (1, 15) y ! 5(3)x

11. (0, 0.1) and (1, 0.5) y ! 0.1(5)x 12. (0, 0.2) and (1, 1.6) y ! 0.2(8)x


13. (3#3$)#3$ 27 14. (x#2$)#7$ x!14"

15. 52#3$ ' 54#3$ 56!3" 16. x3* ) x* x2&


17. 3x # 9 x $ 2 18. 22x ( 3 ! 32 1

19. 49x + x ( " 20. 43x % 2 ! 16

21. 32x ( 5 ! 27x 5 22. 27x ! 32x ( 3 3

4%3

1%2

1&7

1%2

1%3

9&10

x

y

Ox

y

O

1&2



1. y ! 1.5(2)x 2. y ! 4(3)x 3. y ! 3(0.5)x

domain: all real domain: all real domain: all real numbers; range: all numbers; range: all numbers; range: all positive numbers positive numbers positive numbers


4. y ! 5(0.6)x decay 5. y ! 0.1(2)x growth 6. y ! 5 ' 4%x decay

Write an exponential function whose graph passes through the given points.

7. (0, 1) and (%1, 4) 8. (0, 2) and (1, 10) 9. (0, %3) and (1, %1.5)

y ! # $xy ! 2(5)x y ! "3(0.5)x

10. (0, 0.8) and (1, 1.6) 11. (0, %0.4) and (2, %10) 12. (0, *) and (3, 8*)

y ! 0.8(2)x y ! "0.4(5)x y ! &(2)x


13. (2#2$)#8$ 16 14. (n#3$)#75$ n15 15. y#6$ ' y5#6$ y6!6"

16. 13#6$ ' 13#24$ 133!6" 17. n3 ) n* n3 " & 18. 125#11$ ) 5#11$ 52!11"


19. 33x % 5 # 81 x $ 3 20. 76x ! 72x % 20 "5 21. 36n % 5 $ 94n % 3 n $

22. 92x % 1 ! 27x ( 4 14 23. 23n % 1 , ! "nn ) 24. 164n % 1 ! 1282n ( 1

BIOLOGY For Exercises 25 and 26, use the following information.The initial number of bacteria in a culture is 12,000. The number after 3 days is 96,000.

25. Write an exponential function to model the population y of bacteria after x days.y ! 12,000(2)x

26. How many bacteria are there after 6 days? 768,00027. EDUCATION A college with a graduating class of 4000 students in the year 2002

predicts that it will have a graduating class of 4862 in 4 years. Write an exponentialfunction to model the number of students y in the graduating class t years after 2002.y ! 4000(1.05)t

11%2

1%6

1&8

1%2

1%4

x

y

Ox

y

O

Practice (Average)

Exponential Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1

Reading to Learn MathematicsExponential Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1


Less

on

10-

1

Pre-Activity How does an exponential function describe tournament play?

Read the introduction to Lesson 10-1 at the top of page 523 in your textbook.

How many rounds of play would be needed for a tournament with 100players? 7

Reading the Lesson

1. Indicate whether each of the following statements about the exponential function y ! 10x is true or false.

a. The domain is the set of all positive real numbers. false

b. The y-intercept is 1. true

c. The function is one-to-one. true

d. The y-axis is an asymptote of the graph. false

e. The range is the set of all real numbers. false

2. Determine whether each function represents exponential growth or decay.

a. y ! 0.2(3)x. growth b. y ! 3! "x. decay c. y ! 0.4(1.01)x. growth

3. Supply the reason for each step in the following solution of an exponential equation.

92x % 1 ! 27x Original equation

(32)2x % 1 ! (33)x Rewrite each side with a base of 3.32(2x % 1) ! 33x Power of a Power

2(2x % 1) ! 3x Property of Equality for Exponential Functions4x % 2 ! 3x Distributive Propertyx % 2 ! 0 Subtract 3x from each side.

x ! 2 Add 2 to each side.

Helping You Remember

4. One way to remember that polynomial functions and exponential functions are differentis to contrast the polynomial function y ! x2 and the exponential function y ! 2x. Tell atleast three ways they are different.

Sample answer: In y ! x2, the variable x is a base, but in y ! 2x, thevariable x is an exponent. The graph of y ! x2 is symmetric with respectto the y-axis, but the graph of y ! 2x is not. The graph of y ! x2 touchesthe x-axis at (0, 0), but the graph of y ! 2x has the x-axis as an asymptote.You can compute the value of y ! x2 mentally for x ! 100, but you cannotcompute the value of y ! 2x mentally for x ! 100.

2&5


Finding Solutions of xy ! yx

Perhaps you have noticed that if x and y are interchanged in equations suchas x ! y and xy ! 1, the resulting equation is equivalent to the originalequation. The same is true of the equation xy ! yx. However, findingsolutions of xy ! yx and drawing its graph is not a simple process.

Solve each problem. Assume that x and y are positive real numbers.

1. If a # 0, will (a, a) be a solution of xy ! yx? Justify your answer.

2. If c # 0, d # 0, and (c, d) is a solution of xy ! yx, will (d, c) also be a solution? Justify your answer.

3. Use 2 as a value for y in xy ! yx. The equation becomes x2 ! 2x.

a. Find equations for two functions, f(x) and g(x) that you could graph tofind the solutions of x2 ! 2x. Then graph the functions on a separatesheet of graph paper.

b. Use the graph you drew for part a to state two solutions for x2 ! 2x.Then use these solutions to state two solutions for xy ! yx.

4. In this exercise, a graphing calculator will be very helpful. Use the technique of Exercise 3 to complete the tables below. Then graph xy ! yx

for positive values of x and y. If there are asymptotes, show them in yourdiagram using dotted lines. Note that in the table, some values of y callfor one value of x, others call for two.

x

y

O

x y

4

4

5

5

8

8

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1

x y

&12&

&34&

1

2

2

3

3

Study Guide and InterventionLogarithms and Logarithmic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2


Less

on

10-

2

Logarithmic Functions and Expressions

Definition of Logarithm Let b and x be positive numbers, b " 1. The logarithm of x with base b is denoted with Base b logb x and is defined as the exponent y that makes the equation by ! x true.

The inverse of the exponential function y ! bx is the logarithmic function x ! by.This function is usually written as y ! logb x.

1. The function is continuous and one-to-one.

Properties of2. The domain is the set of all positive real numbers.

Logarithmic Functions3. The y-axis is an asymptote of the graph.4. The range is the set of all real numbers.5. The graph contains the point (0, 1).

Write an exponential equation equivalent to log3 243 ! 5.35 ! 243

Write a logarithmic equation equivalent to 6"3 ! .

log6 ! %3

Evaluate log8 16.

8&43

&

! 16, so log8 16 ! .

Write each equation in logarithmic form.

1. 27 ! 128 2. 3%4 ! 3. ! "3!

log2 128 ! 7 log3 ! "4 log%17

% ! 3

Write each equation in exponential form.

4. log15 225 ! 2 5. log3 ! %3 6. log4 32 !

152 ! 225 3"3 ! 4%52

%! 32

Evaluate each expression.

7. log4 64 3 8. log2 64 6 9. log100 100,000 2.5

10. log5 625 4 11. log27 81 12. log25 5

13. log2 "7 14. log10 0.00001 "5 15. log4 "2.51&32

1&128

1%2

4%3

1%27

5&2

1&27

1%343

1%81

1&343

1&7

1&81

4&3

1&216

1%216

Example 1Example 1

Example 2Example 2

Example 3Example 3

ExercisesExercises

Brad and Anissa Stuckey

All


Solve Logarithmic Equations and Inequalities

Logarithmic to If b # 1, x # 0, and logb x # y, then x # by.Exponential Inequality If b # 1, x # 0, and logb x $ y, then 0 $ x $ by.

Property of Equality for If b is a positive number other than 1, Logarithmic Functions then logb x ! logb y if and only if x ! y.

Property of Inequality for If b # 1, then logb x # logb y if and only if x # y, Logarithmic Functions and logb x $ logb y if and only if x $ y.


Logarithms and Logarithmic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2

Solve log2 2x ! 3.log2 2x ! 3 Original equation

2x ! 23 Definition of logarithm

2x ! 8 Simplify.

x ! 4 Simplify.

The solution is x ! 4.

Solve log5 (4x " 3) ' 3.log5 (4x % 3) $ 3 Original equation

0 $ 4x % 3 $ 53 Logarithmic to exponential inequality

3 $ 4x $ 125 ( 3 Addition Property of Inequalities

$ x $ 32 Simplify.

The solution set is 'x | $ x $ 32(.3&4

3&4


ExercisesExercises

Solve each equation or inequality.

1. log2 32 ! 3x 2. log3 2c ! %2

3. log2x 16 ! %2 4. log25 ! " ! 10

5. log4 (5x ( 1) ! 2 3 6. log8 (x % 5) ! 9

7. log4 (3x % 1) ! log4 (2x ( 3) 4 8. log2 (x2 % 6) ! log2 (2x ( 2) 4

9. logx ( 4 27 ! 3 "1 10. log2 (x (3) ! 4 13

11. logx 1000 ! 3 10 12. log8 (4x ( 4) ! 2 15

13. log2 2x # 2 x $ 2 14. log5 x # 2 x $ 25

15. log2 (3x ( 1) $ 4 " ' x ' 5 16. log4 (2x) # % x $

17. log3 (x ( 3) $ 3 "3 ' x ' 24 18. log27 6x # x $ 3%2

2&3

1%4

1&2

1%3

2&3

1&2

x&2

1%8

1%18

5%3

Brad and Anissa Stuckey

Odds

Skills PracticeLogarithms and Logarithmic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2


Less

on

10-

2


1. 23 ! 8 log2 8 ! 3 2. 32 ! 9 log3 9 ! 2

3. 8%2 ! log8 ! "2 4. ! "2! log%

13

% ! 2


5. log3 243 ! 5 35 ! 243 6. log4 64 ! 3 43 ! 64

7. log9 3 ! 9%12

%! 3 8. log5 ! %2 5"2 !


9. log5 25 2 10. log9 3

11. log10 1000 3 12. log125 5

13. log4 "3 14. log5 "4

15. log8 83 3 16. log27 "

Solve each equation or inequality. Check your solutions.

17. log3 x ! 5 243 18. log2 x ! 3 8

19. log4 y $ 0 0 ' y ' 1 20. log&14

& x ! 3

21. log2 n # %2 n $ 22. logb 3 ! 9

23. log6 (4x ( 12) ! 2 6 24. log2 (4x % 4) # 5 x $ 9

25. log3 (x ( 2) ! log3 (3x) 1 26. log6 (3y % 5) , log6 (2y ( 3) y ) 8

1&2

1%4

1%64

1%3

1&3

1&625

1&64

1%3

1%2

1%25

1&25

1&2

1%9

1&9

1&3

1%64

1&64



1. 53 ! 125 log5 125 ! 3 2. 70 ! 1 log7 1 ! 0 3. 34 ! 81 log3 81 ! 4

4. 3%4 ! 5. ! "3! 6. 7776

&15

&

! 6

log3 ! "4 log%14

% ! 3 log7776 6 !


7. log6 216 ! 3 63 ! 216 8. log2 64 ! 6 26 ! 64 9. log3 ! %4 3"4 !

10. log10 0.00001 ! %5 11. log25 5 ! 12. log32 8 !

10"5 ! 0.00001 25%12

%! 5 32

%35

%! 8


13. log3 81 4 14. log10 0.0001 "4 15. log2 "4 16. log&13

& 27 "3

17. log9 1 0 18. log8 4 19. log7 "2 20. log6 64 4

21. log3 "1 22. log4 "4 23. log9 9(n ( 1) n # 1 24. 2log2 32 32

Solve each equation or inequality. Check your solutions.

25. log10 n ! %3 26. log4 x # 3 x $ 64 27. log4 x ! 8

28. log&15

& x ! %3 125 29. log7 q $ 0 0 ' q ' 1 30. log6 (2y ( 8) , 2 y ) 14

31. logy 16 ! %4 32. logn ! %3 2 33. logb 1024 ! 5 4

34. log8 (3x ( 7) $ log8 (7x ( 4) 35. log7 (8x ( 20) ! log7 (x ( 6) 36. log3 (x2 % 2) ! log3 x

x $ "2 2

37. SOUND Sounds that reach levels of 130 decibels or more are painful to humans. Whatis the relative intensity of 130 decibels? 1013

38. INVESTING Maria invests $1000 in a savings account that pays 8% interestcompounded annually. The value of the account A at the end of five years can bedetermined from the equation log A ! log[1000(1 ( 0.08)5]. Find the value of A to thenearest dollar. $1469

3%4

1&8

1%2

3&2

1%1000

1&256

1&3

1&49

2%3

1&16

3&5

1&2

1%81

1&81

1%5

1%64

1%81

1&64

1&4

1&81

Practice (Average)

Logarithms and Logarithmic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2

Reading to Learn MathematicsLogarithms and Logarithmic Functions

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Pre-Activity Why is a logarithmic scale used to measure sound?


How many times louder than a whisper is normal conversation?104 or 10,000 times

Reading the Lesson1. a. Write an exponential equation that is equivalent to log3 81 ! 4. 34 ! 81

b. Write a logarithmic equation that is equivalent to 25%&12

&! . log25 ! "

c. Write an exponential equation that is equivalent to log4 1 ! 0. 40 ! 1

d. Write a logarithmic equation that is equivalent to 10%3 ! 0.001. log10 0.001 ! "3

e. What is the inverse of the function y ! 5x? y ! log5 x

f. What is the inverse of the function y ! log10 x? y ! 10x

2. Match each function with its graph.

a. y ! 3x IV b. y ! log3 x I c. y ! ! "xII

I. II. III.

3. Indicate whether each of the following statements about the exponential function y ! log5 x is true or false.

a. The y-axis is an asymptote of the graph. trueb. The domain is the set of all real numbers. falsec. The graph contains the point (5, 0). falsed. The range is the set of all real numbers. truee. The y-intercept is 1. false

Helping You Remember4. An important skill needed for working with logarithms is changing an equation between

logarithmic and exponential forms. Using the words base, exponent, and logarithm, describean easy way to remember and apply the part of the definition of logarithm that says,“logb x ! y if and only if by ! x.” Sample answer: In these equations, b standsfor base. In log form, b is the subscript, and in exponential form, b is thenumber that is raised to a power. A logarithm is an exponent, so y, which isthe log in the first equation, becomes the exponent in the second equation.

x

y

Ox

y

O

x

y

O

1&3

1%2

1%5

1&5


Enrichment

NAME ______________________________________________ DATE______________ PERIOD _____

10-210-2

Musical RelationshipsThe frequencies of notes in a musical scale that are one octave apart arerelated by an exponential equation. For the eight C notes on a piano, theequation is Cn ! C12n " 1, where Cn represents the frequency of note Cn.

1. Find the relationship between C1 and C2.

2. Find the relationship between C1 and C4.

The frequencies of consecutive notes are related by a common ratio r. The general equation is fn ! f1rn " 1.

3. If the frequency of middle C is 261.6 cycles per second and the frequency of the next higher C is 523.2 cycles per second, find the common ratio r. (Hint: The two C’s are 12 notes apart.) Write the answer as a radicalexpression.

4. Substitute decimal values for r and f1 to find a specific equation for fn.

5. Find the frequency of F# above middle C.

6. Frets are a series of ridges placed across the fingerboard of a guitar. Theyare spaced so that the sound made by pressing a string against one frethas about 1.0595 times the wavelength of the sound made by using thenext fret. The general equation is wn ! w0(1.0595)n. Describe thearrangement of the frets on a guitar.

Study Guide and InterventionProperties of Logarithms

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Properties of Logarithms Properties of exponents can be used to develop thefollowing properties of logarithms.

Product Property For all positive numbers m, n, and b, where b # 1, of Logarithms logb mn ! logb m $ logb n.

Quotient Property For all positive numbers m, n, and b, where b # 1, of Logarithms logb %

mn% ! logb m " logb n.

Power Property For any real number p and positive numbers m and b, of Logarithms where b # 1, logb mp ! p logb m.

Use log3 28 ! 3.0331 and log3 4 ! 1.2619 to approximate the value of each expression.ExampleExample

a. log3 36

log3 36 ! log3 (32 & 4)! log3 32 $ log3 4! 2 $ log3 4! 2 $ 1.2619! 3.2619

b. log3 7

log3 7 ! log3 " #! log3 28 " log3 4! 3.0331 " 1.2619! 1.7712

c. log3 256

log3 256 ! log3 (44)! 4 & log3 4! 4(1.2619)! 5.0476

28%4

ExercisesExercises

Use log12 3 ! 0.4421 and log12 7 ! 0.7831 to evaluate each expression.

1. log12 21 1.2252 2. log12 0.3410 3. log12 49 1.5662

4. log12 36 1.4421 5. log12 63 1.6673 6. log12 !0.2399

7. log12 0.2022 8. log12 16,807 3.9155 9. log12 441 2.4504

Use log5 3 ! 0.6826 and log5 4 ! 0.8614 to evaluate each expression.

10. log5 12 1.5440 11. log5 100 2.8614 12. log5 0.75 !0.1788

13. log5 144 3.0880 14. log5 0.3250 15. log5 375 3.6826

16. log5 1.3$ 0.1788 17. log5 !0.3576 18. log5 1.730481%5

9%16

27%16

81%49

27%49

7%3


Solve Logarithmic Equations You can use the properties of logarithms to solveequations involving logarithms.

Solve each equation.

a. 2 log3 x ! log3 4 " log3 25

2 log3 x ! log3 4 " log3 25 Original equation

log3 x2 ! log3 4 " log3 25 Power Property

log3 " log3 25 Quotient Property

" 25 Property of Equality for Logarithmic Functions

x2 " 100 Multiply each side by 4.

x " #10 Take the square root of each side.

Since logarithms are undefined for x $ 0, !10 is an extraneous solution.The only solution is 10.

b. log2 x % log2 (x % 2) " 3

log2 x % log2 (x % 2) " 3 Original equation

log2 x(x % 2) " 3 Product Property

x(x % 2) " 23 Definition of logarithm

x2 % 2x " 8 Distributive Property

x2 % 2x ! 8 " 0 Subtract 8 from each side.

(x % 4)(x ! 2) " 0 Factor.

x " 2 or x " !4 Zero Product Property

Since logarithms are undefined for x $ 0, !4 is an extraneous solution.The only solution is 2.

Solve each equation. Check your solutions.

1. log5 4 % log5 2x " log5 24 3 2. 3 log4 6 ! log4 8 " log4 x 27

3. log6 25 % log6 x " log6 20 4 4. log2 4 ! log2 (x % 3) " log2 8 !

5. log6 2x ! log6 3 " log6 (x ! 1) 3 6. 2 log4 (x % 1) " log4 (11 ! x) 2

7. log2 x ! 3 log2 5 " 2 log2 10 12,500 8. 3 log2 x ! 2 log2 5x " 2 100

9. log3 (c % 3) ! log3 (4c ! 1) " log3 5 10. log5 (x % 3) ! log5 (2x ! 1) " 24"7

8"19

5"2

1&2

x2&4

x2&4


Properties of Logarithms

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10-310-3

ExampleExample

ExercisesExercises

Skills PracticeProperties of Logarithms

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Use log2 3 ! 1.5850 and log2 5 ! 2.3219 to approximate the value of eachexpression.

1. log2 25 4.6438 2. log2 27 4.755

3. log2 !0.7369 4. log2 0.7369

5. log2 15 3.9069 6. log2 45 5.4919

7. log2 75 6.2288 8. log2 0.6 !0.7369

9. log2 !1.5850 10. log2 0.8481


11. log10 27 " 3 log10 x 3 12. 3 log7 4 " 2 log7 b 8

13. log4 5 % log4 x " log4 60 12 14. log6 2c % log6 8 " log6 80 5

15. log5 y ! log5 8 " log5 1 8 16. log2 q ! log2 3 " log2 7 21

17. log9 4 % 2 log9 5 " log9 w 100 18. 3 log8 2 ! log8 4 " log8 b 2

19. log10 x % log10 (3x ! 5) " log10 2 2 20. log4 x % log4 (2x ! 3) " log4 2 2

21. log3 d % log3 3 " 3 9 22. log10 y ! log10 (2 ! y) " 0 1

23. log2 s % 2 log2 5 " 0 24. log2 (x % 4) ! log2 (x ! 3) " 3 4

25. log4 (n % 1) ! log4 (n ! 2) " 1 3 26. log5 10 % log5 12 " 3 log5 2 % log5 a 15

1"25

9&5

1&3

5&3

3&5


Use log10 5 ! 0.6990 and log10 7 ! 0.8451 to approximate the value of eachexpression.

1. log10 35 1.5441 2. log10 25 1.3980 3. log10 0.1461 4. log10 !0.1461

5. log10 245 2.3892 6. log10 175 2.2431 7. log10 0.2 !0.6990 8. log10 0.5529


9. log7 n ! log7 8 4 10. log10 u ! log10 4 8

11. log6 x " log6 9 ! log6 54 6 12. log8 48 # log8 w ! log8 4 12

13. log9 (3u " 14) # log9 5 ! log9 2u 2 14. 4 log2 x " log2 5 ! log2 405 3

15. log3 y ! #log3 16 " log3 64 16. log2 d ! 5 log2 2 # log2 8 4

17. log10 (3m # 5) " log10 m ! log10 2 2 18. log10 (b " 3) " log10 b ! log10 4 1

19. log8 (t " 10) # log8 (t # 1) ! log8 12 2 20. log3 (a " 3) " log3 (a " 2) ! log3 6 0

21. log10 (r " 4) # log10 r ! log10 (r " 1) 2 22. log4 (x2 # 4) # log4 (x " 2) ! log4 1 3

23. log10 4 " log10 w ! 2 25 24. log8 (n # 3) " log8 (n " 4) ! 1 4

25. 3 log5 (x2 " 9) # 6 ! 0 "4 26. log16 (9x " 5) # log16 (x2 # 1) ! 3

27. log6 (2x # 5) " 1 ! log6 (7x " 10) 8 28. log2 (5y " 2) # 1 ! log2 (1 # 2y) 0

29. log10 (c2 # 1) # 2 ! log10 (c " 1) 101 30. log7 x " 2 log7 x # log7 3 ! log7 72 6

31. SOUND The loudness L of a sound in decibels is given by L ! 10 log10 R, where R is thesound’s relative intensity. If the intensity of a certain sound is tripled, by how manydecibels does the sound increase? about 4.8 db

32. EARTHQUAKES An earthquake rated at 3.5 on the Richter scale is felt by many people,and an earthquake rated at 4.5 may cause local damage. The Richter scale magnitudereading m is given by m ! log10 x, where x represents the amplitude of the seismic wavecausing ground motion. How many times greater is the amplitude of an earthquake thatmeasures 4.5 on the Richter scale than one that measures 3.5? 10 times

1$2

1#4

1$3

3$2

2$3

25$7

5$7

7$5

Practice (Average)

Properties of Logarithms

NAME ______________________________________________ DATE______________ PERIOD _____

10-310-3

Reading to Learn MathematicsProperties of Logarithms

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Pre-Activity How are the properties of exponents and logarithms related?

Read the introduction to Lesson 10-3 at the top of page 541 in your textbook.Find the value of log5 125. 3 Find the value of log5 5. 1Find the value of log5 (125 % 5). 2Which of the following statements is true? BA. log5 (125 % 5) ! (log5 125) % (log5 5)B. log5 (125 % 5) ! log5 125 # log5 5

Reading the Lesson1. Each of the properties of logarithms can be stated in words or in symbols. Complete the

statements of these properties in words.

a. The logarithm of a quotient is the of the logarithms of the

and the .

b. The logarithm of a power is the of the logarithm of the base and

the .

c. The logarithm of a product is the of the logarithms of its

.

2. State whether each of the following equations is true or false. If the statement is true,name the property of logarithms that is illustrated.

a. log3 10 ! log3 30 # log3 3 true; Quotient Propertyb. log4 12 ! log4 4 " log4 8 falsec. log2 81 ! 2 log2 9 true; Power Propertyd. log8 30 ! log8 5 & log8 6 false

3. The algebraic process of solving the equation log2 x " log2 (x " 2) ! 3 leads to “x ! #4or x ! 2.” Does this mean that both #4 and 2 are solutions of the logarithmic equation?Explain your reasoning. Sample answer: No; 2 is a solution because it checks: log2 2 $ log2 (2 $ 2) % log2 2 $ log2 4 % 1 $ 2 % 3. However,because log2 (!4) and log2 (! 2) are undefined, !4 is an extraneoussolution and must be eliminated. The only solution is 2.

Helping You Remember4. A good way to remember something is to relate it something you already know. Use words

to explain how the Product Property for exponents can help you remember the productproperty for logarithms. Sample answer: When you multiply two numbers orexpressions with the same base, you add the exponents and keep thesame base. Logarithms are exponents, so to find the logarithm of aproduct, you add the logarithms of the factors, keeping the same base.

factorssum

exponentproduct

denominatornumeratordifference


SpiralsConsider an angle in standard position with its vertex at a point O called thepole. Its initial side is on a coordinatized axis called the polar axis. A point Pon the terminal side of the angle is named by the polar coordinates (r, !),where r is the directed distance of the point from O and ! is the measure ofthe angle. Graphs in this system may be drawn on polar coordinate papersuch as the kind shown below.

1. Use a calculator to complete the table for log2r " #12!0#.

(Hint: To find ! on a calculator, press 120 r 2 .)

2. Plot the points found in Exercise 1 on the grid above and connect to form a smooth curve.

This type of spiral is called a logarithmic spiral because the angle measures are proportional to the logarithms of the radii.

r 1 2 3 4 5 6 7 8

) LOG!) LOG"

0

10

20

30

40

5060

708090100110

120130

140

150

160

170

180

190

200

210

220

230240

250260 270 280

290300

310

320

330

340

350

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

10-310-3

Study Guide and InterventionCommon Logarithms

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10-410-4


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Common Logarithms Base 10 logarithms are called common logarithms. Theexpression log10 x is usually written without the subscript as log x. Use the key onyour calculator to evaluate common logarithms.The relation between exponents and logarithms gives the following identity.

Inverse Property of Logarithms and Exponents 10log x ! x

Evaluate log 50 to four decimal places.Use the LOG key on your calculator. To four decimal places, log 50 ! 1.6990.

Solve 32x ! 1 " 12.32x " 1 ! 12 Original equation

log 32x " 1 ! log 12 Property of Equality for Logarithms

(2x " 1) log 3 ! log 12 Power Property of Logarithms

2x " 1 ! Divide each side by log 3.

2x ! # 1 Subtract 1 from each side.

x ! ! # 1" Multiply each side by .

x # 0.6309

Use a calculator to evaluate each expression to four decimal places.

1. log 18 2. log 39 3. log 1201.2553 1.5911 2.0792

4. log 5.8 5. log 42.3 6. log 0.0030.7634 1.6263 #2.5229

Solve each equation or inequality. Round to four decimal places.

7. 43x ! 12 0.5975 8. 6x " 2 ! 18 #0.3869

9. 54x # 2 ! 120 1.2437 10. 73x # 1 $ 21 {x |x $ 0.8549}

11. 2.4x " 4 ! 30 #0.1150 12. 6.52x $ 200 {x |x $ 1.4153}

13. 3.64x # 1 ! 85.4 1.1180 14. 2x " 5 ! 3x # 2 13.9666

15. 93x ! 45x " 2 #8.1595 16. 6x # 5 ! 27x " 3 #3.6069

1%2

log 12%log 3

1%2

log 12%log 3

log 12%log 3

LOG

ExercisesExercises

Example 1Example 1

Example 2Example 2


Change of Base Formula The following formula is used to change expressions withdifferent logarithmic bases to common logarithm expressions.

Change of Base Formula For all positive numbers a, b, and n, where a & 1 and b & 1, loga n !

Express log8 15 in terms of common logarithms. Then approximateits value to four decimal places.

log8 15 ! Change of Base Formula

# 1.3023 Simplify.

The value of log8 15 is approximately 1.3023.

Express each logarithm in terms of common logarithms. Then approximate itsvalue to four decimal places.

1. log3 16 2. log2 40 3. log5 35

, 2.5237 , 5.3219 , 2.2091

4. log4 22 5. log12 200 6. log2 50

, 2.2297 , 2.1322 , 5.6439

7. log5 0.4 8. log3 2 9. log4 28.5

, #0.5693 , 0.6309 , 2.4164

10. log3 (20)2 11. log6 (5)4 12. log8 (4)5

, 5.4537 , 3.5930 , 3.3333

13. log5 (8)3 14. log2 (3.6)6 15. log12 (10.5)4

, 3.8761 , 11.0880 , 3.7851

16. log3 $150% 17. log43$39% 18. log5

4$1600%

, 2.2804 , 0.8809 , 1.1460log 1600%%4 log 5

log 39%3 log 4

log 150%2 log 3

4 log 10.5%%log 12

6 log 3.6%%log 2

3 log 8%log 5

5 log 4%log 8

4 log 5%log 6

2 log 20%%log 3

log 28.5%%log 4

log 2%log 3

log 0.4%log 5

log 50%log 2

log 200%log 12

log 22%log 4

log 35%log 5

log 40%log 2

log 16%log 3

log10 15%log10 8

logb n%logb a


Common Logarithms

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ExampleExample

ExercisesExercises

Skills PracticeCommon Logarithms

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1. log 6 0.7782 2. log 15 1.1761

3. log 1.1 0.0414 4. log 0.3 #0.5229

Use the formula pH " #log[H!] to find the pH of each substance given itsconcentration of hydrogen ions.

5. gastric juices: [H"] ! 1.0 ' 10#1 mole per liter 1.0

6. tomato juice: [H"] ! 7.94 ' 10#5 mole per liter 4.1

7. blood: [H"] ! 3.98 ' 10#8 mole per liter 7.4

8. toothpaste: [H"] ! 1.26 ' 10#10 mole per liter 9.9


9. 3x ( 243 {x |x & 5} 10. 16v ) !v "v ' # #11. 8p ! 50 1.8813 12. 7y ! 15 1.3917

13. 53b ! 106 0.9659 14. 45k ! 37 0.5209

15. 127p ! 120 0.2752 16. 92m ! 27 0.75

17. 3r # 5 ! 4.1 6.2843 18. 8y " 4 ( 15 {y |y & #2.6977}

19. 7.6d " 3 ! 57.2 #1.0048 20. 0.5t # 8 ! 16.3 3.9732

21. 42x2! 84 (1.0888 22. 5x2 " 1! 10 (0.6563


23. log3 7 ; 1.7712 24. log5 66 ; 2.6032

25. log2 35 ; 5.1293 26. log6 10 ; 1.2851log10 10%%log10 6

log10 35%%log10 2

log10 66%%log10 5

log10 7%log10 3

1%2

1%4



1. log 101 2.0043 2. log 2.2 0.3424 3. log 0.05 #1.3010

Use the formula pH " #log[H!] to find the pH of each substance given itsconcentration of hydrogen ions.

4. milk: [H"] ! 2.51 ' 10#7 mole per liter 6.6

5. acid rain: [H"] ! 2.51 ' 10#6 mole per liter 5.6

6. black coffee: [H"] ! 1.0 ' 10#5 mole per liter 5.0

7. milk of magnesia: [H"] ! 3.16 ' 10#11 mole per liter 10.5


8. 2x * 25 {x |x ) 4.6439} 9. 5a ! 120 2.9746 10. 6z ! 45.6 2.1319

11. 9m $ 100 {m |m $ 2.0959} 12. 3.5x ! 47.9 3.0885 13. 8.2y ! 64.5 1.9802

14. 2b " 1 ) 7.31 {b |b ' 1.8699} 15. 42x ! 27 1.1887 16. 2a # 4 ! 82.1 10.3593

17. 9z # 2 ( 38 {z |z & 3.6555} 18. 5w " 3 ! 17 #1.2396 19. 30x2! 50 (1.0725

20. 5x2 # 3 ! 72 (2.3785 21. 42x ! 9x " 1 3.8188 22. 2n " 1 ! 52n # 1 0.9117


23. log5 12 ; 1.5440 24. log8 32 ; 1.6667 25. log11 9 ; 0.9163

26. log2 18 ; 4.1699 27. log9 6 ; 0.8155 28. log7 $8% ;

29. HORTICULTURE Siberian irises flourish when the concentration of hydrogen ions [H"]in the soil is not less than 1.58 ' 10#8 mole per liter. What is the pH of the soil in whichthese irises will flourish? 7.8 or less

30. ACIDITY The pH of vinegar is 2.9 and the pH of milk is 6.6. How many times greater isthe hydrogen ion concentration of vinegar than of milk? about 5000

31. BIOLOGY There are initially 1000 bacteria in a culture. The number of bacteria doubleseach hour. The number of bacteria N present after t hours is N ! 1000(2) t. How long willit take the culture to increase to 50,000 bacteria? about 5.6 h

32. SOUND An equation for loudness L in decibels is given by L ! 10 log R, where R is thesound’s relative intensity. An air-raid siren can reach 150 decibels and jet engine noisecan reach 120 decibels. How many times greater is the relative intensity of the air-raidsiren than that of the jet engine noise? 1000

log10 8%2 log10 7

log10 6%%log10 9

log10 18%%log10 2

log10 9%%log10 11

log10 32%%log10 8

log10 12%%log10 5

Practice (Average)

Common Logarithms

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10-410-4

0.5343

Reading to Learn MathematicsCommon Logarithms

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10-410-4


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Pre-Activity Why is a logarithmic scale used to measure acidity?


Which substance is more acidic, milk or tomatoes? tomatoes

Reading the Lesson

1. Rhonda used the following keystrokes to enter an expression on her graphing calculator:

17

The calculator returned the result 1.230448921.Which of the following conclusions are correct? a, c, and d

a. The base 10 logarithm of 17 is about 1.2304.

b. The base 17 logarithm of 10 is about 1.2304.

c. The common logarithm of 17 is about 1.230449.

d. 101.230448921 is very close to 17.

e. The common logarithm of 17 is exactly 1.230448921.

2. Match each expression from the first column with an expression from the second columnthat has the same value.

a. log2 2 iv i. log4 1

b. log 12 iii ii. log2 8

c. log3 1 i iii. log10 12

d. log5 v iv. log5 5

e. log 1000 ii v. log 0.1

3. Calculators do not have keys for finding base 8 logarithms directly. However, you can use

a calculator to find log8 20 if you apply the formula.

Which of the following expressions are equal to log8 20? B and C

A. log20 8 B. C. D.


4. Sometimes it is easier to remember a formula if you can state it in words. State thechange of base formula in words. Sample answer: To change the logarithm of anumber from one base to another, divide the log of the original numberin the old base by the log of the new base in the old base.

log 8!log 20

log 20!log 8

log10 20!log10 8

change of base

1!5

ENTER) LOG


The Slide RuleBefore the invention of electronic calculators, computations were oftenperformed on a slide rule. A slide rule is based on the idea of logarithms. It hastwo movable rods labeled with C and D scales. Each of the scales is logarithmic.

To multiply 2 ! 3 on a slide rule, move the C rod to the right as shownbelow. You can find 2 ! 3 by adding log 2 to log 3, and the slide rule adds thelengths for you. The distance you get is 0.778, or the logarithm of 6.

Follow the steps to make a slide rule.

1. Use graph paper that has small squares, such as 10 squares to the inch. Using the scales shown at the right, plot the curve y " log x for x " 1, 1.5,and the whole numbers from 2 through 10. Make an obvious heavy dot for each point plotted.

2. You will need two strips of cardboard. A 5-by-7 index card, cut in half the long way,will work fine. Turn the graph you made in Exercise 1 sideways and use it to marka logarithmic scale on each of the twostrips. The figure shows the mark for 2 being drawn.

3. Explain how to use a slide rule to divide 8 by 2.

0

0.1

0.2

0.3 y

12

1 1.5 2

y = log x

0.1

0.2

1 2

1

21

CD

2

4

3

6

4 5 6 7 8 9

83 5 7 9

log 6

log 3log 2

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

C

D

Enrichment

NAME ______________________________________________ DATE______________ PERIOD _____

10-410-4

Study Guide and InterventionBase e and Natural Logarithms

NAME ______________________________________________ DATE______________ PERIOD _____

10-510-5


Less

on

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Base e and Natural Logarithms The irrational number e ! 2.71828… often occursas the base for exponential and logarithmic functions that describe real-world phenomena.

Natural Base e As n increases, "1 # #napproaches e ! 2.71828….

ln x " loge x

The functions y " ex and y " ln x are inverse functions.

Inverse Property of Base e and Natural Logarithms eln x " x ln ex " x

Natural base expressions can be evaluated using the ex and ln keys on your calculator.

Evaluate ln 1685.Use a calculator.ln 1685 ! 7.4295

Write a logarithmic equation equivalent to e2x ! 7.e2x " 7 → loge 7 " 2x or 2x " ln 7

Evaluate ln e18.Use the Inverse Property of Base e and Natural Logarithms.ln e18 " 18


1. ln 732 2. ln 84,350 3. ln 0.735 4. ln 1006.5958 11.3427 "0.3079 4.6052

5. ln 0.0824 6. ln 2.388 7. ln 128,245 8. ln 0.00614"2.4962 0.8705 11.7617 "5.0929

Write an equivalent exponential or logarithmic equation.

9. e15 " x 10. e3x " 45 11. ln 20 " x 12. ln x " 8ln x ! 15 3x ! ln 45 ex ! 20 x ! e8

13. e$5x " 0.2 14. ln (4x) " 9.6 15. e8.2 " 10x 16. ln 0.0002 " x"5x ! ln 0.2 4x ! e9.6 ln 10x ! 8.2 ex ! 0.0002


17. ln e3 18. eln 42 19. eln 0.5 20. ln e16.2

3 42 0.5 16.2

1%n

Example 1Example 1

Example 2Example 2

Example 3Example 3

ExercisesExercises


Equations and Inequalities with e and ln All properties of logarithms fromearlier lessons can be used to solve equations and inequalities with natural logarithms.


a. 3e2x # 2 " 103e2x # 2 " 10 Original equation

3e2x " 8 Subtract 2 from each side.

e2x " Divide each side by 3.

ln e2x " ln Property of Equality for Logarithms

2x " ln Inverse Property of Exponents and Logarithms

x " ln Multiply each side by %12%.

x ! 0.4904 Use a calculator.

b. ln (4x $ 1) & 2

ln (4x $ 1) & 2 Original inequality

eln (4x $ 1) & e2 Write each side using exponents and base e.

0 & 4x $ 1 & e2 Inverse Property of Exponents and Logarithms

1 & 4x & e2 # 1 Addition Property of Inequalities

& x & (e2 # 1) Multiplication Property of Inequalities

0.25 & x & 2.0973 Use a calculator.


1. e4x " 120 2. ex ' 25 3. ex $ 2 # 4 " 211.1969 {x|x # 3.2189} 4.8332

4. ln 6x ( 4 5. ln (x # 3) $ 5 " $2 6. e$8x ' 50x $ 9.0997 17.0855 {x |x $ "0.4890}

7. e4x $ 1 $ 3 " 12 8. ln (5x # 3) " 3.6 9. 2e3x # 5 " 20.9270 6.7196 no solution

10. 6 # 3ex # 1 " 21 11. ln (2x $ 5) " 8 12. ln 5x # ln 3x ) 90.6094 1492.9790 {x |x % 23.2423}

1%4

1%4

8%3

1%2

8%3

8%3

8%3


Base e and Natural Logarithms

NAME ______________________________________________ DATE______________ PERIOD _____

10-510-5

ExampleExample

ExercisesExercises

NAME ______________________________________________ DATE______________ PERIOD _____

10-510-5


Less

on

10-

5


1. e3 20.0855 2. e$2 0.1353

3. ln 2 0.6931 4. ln 0.09 "2.4079


5. ex " 3 x ! ln 3 6. e4 " 8x 4 ! ln 8x

7. ln 15 " x ex ! 15 8. ln x ! 0.6931 x ! e0.6931


9. eln 3 3 10. eln 2x 2x

11. ln e$2.5 "2.5 12. ln ey y


13. ex ( 5 {x |x $ 1.6094} 14. ex & 3.2 {x |x & 1.1632}

15. 2ex $ 1 " 11 1.7918 16. 5ex # 3 " 18 1.0986

17. e3x " 30 1.1337 18. e$4x ) 10 {x |x & "0.5756}

19. e5x # 4 ) 34 {x |x % 0.6802} 20. 1 $ 2e2x " $19 1.1513

21. ln 3x " 2 2.4630 22. ln 8x " 3 2.5107

23. ln (x $ 2) " 2 9.3891 24. ln (x # 3) " 1 "0.2817

25. ln (x # 3) " 4 51.5982 26. ln x # ln 2x " 2 1.9221

Skills PracticeBase e and Natural Logarithms



1. e1.5 4.4817 2. ln 8 2.0794 3. ln 3.2 1.1632 4. e$0.6 0.5488

5. e4.2 66.6863 6. ln 1 0 7. e$2.5 0.0821 8. ln 0.037 "3.2968


9. ln 50 " x 10. ln 36 " 2x 11. ln 6 ! 1.7918 12. ln 9.3 ! 2.2300

ex ! 50 e2x ! 36 e1.7918 ! 6 e2.2300 ! 9.3

13. ex " 8 14. e5 " 10x 15. e$x " 4 16. e2 " x # 1

x ! ln 8 5 ! ln 10x x ! "ln 4 2 ! ln (x ' 1)


17. eln 12 12 18. eln 3x 3x 19. ln e$1 "1 20. ln e$2y "2y


21. ex & 9 22. e$x " 31 23. ex " 1.1 24. ex " 5.8

{x |x & 2.1972} "3.4340 0.0953 1.7579

25. 2ex $ 3 " 1 26. 5ex # 1 ( 7 27. 4 # ex " 19 28. $3ex # 10 & 8

0.6931 {x |x $ 0.1823} 2.7081 {x |x % "0.4055}

29. e3x " 8 30. e$4x " 5 31. e0.5x " 6 32. 2e5x " 24

0.6931 "0.4024 3.5835 0.4970

33. e2x # 1 " 55 34. e3x $ 5 " 32 35. 9 # e2x " 10 36. e$3x # 7 ( 15

1.9945 1.2036 0 {x |x # "0.6931}

37. ln 4x " 3 38. ln ($2x) " 7 39. ln 2.5x " 10 40. ln (x $ 6) " 1

5.0214 "548.3166 8810.5863 8.7183

41. ln (x # 2) " 3 42. ln (x # 3) " 5 43. ln 3x # ln 2x " 9 44. ln 5x # ln x " 7

18.0855 145.4132 36.7493 14.8097

INVESTING For Exercises 45 and 46, use the formula for continuouslycompounded interest, A ! Pert, where P is the principal, r is the annual interestrate, and t is the time in years.

45. If Sarita deposits $1000 in an account paying 3.4% annual interest compoundedcontinuously, what is the balance in the account after 5 years? $1185.30

46. How long will it take the balance in Sarita’s account to reach $2000? about 20.4 yr

47. RADIOACTIVE DECAY The amount of a radioactive substance y that remains after t years is given by the equation y " aekt, where a is the initial amount present and k isthe decay constant for the radioactive substance. If a " 100, y " 50, and k " $0.035,find t. about 19.8 yr

Practice (Average)

Base e and Natural Logarithms

NAME ______________________________________________ DATE______________ PERIOD _____

10-510-5

Reading to Learn MathematicsBase e and Natural Logarithms

NAME ______________________________________________ DATE ____________ PERIOD _____

10-510-5


Less

on

10-

5

Pre-Activity How is the natural base e used in banking?


Suppose that you deposit $675 in a savings account that pays an annualinterest rate of 5%. In each case listed below, indicate which method ofcompounding would result in more money in your account at the end of oneyear.a. annual compounding or monthly compounding monthlyb. quarterly compounding or daily compounding dailyc. daily compounding or continuous compounding continuous

Reading the Lesson1. Jagdish entered the following keystrokes in his calculator:

5

The calculator returned the result 1.609437912. Which of the following conclusions arecorrect? d and fa. The common logarithm of 5 is about 1.6094.

b. The natural logarithm of 5 is exactly 1.609437912.

c. The base 5 logarithm of e is about 1.6094.

d. The natural logarithm of 5 is about 1.609438.

e. 101.609437912 is very close to 5.

f. e1.609437912 is very close to 5.

2. Match each expression from the first column with its value in the second column. Somechoices may be used more than once or not at all.

a. eln 5 IV I. 1

b. ln 1 V II. 10

c. eln e VI III. !1

d. ln e5 IV IV. 5

e. ln e I V. 0

f. ln ! " III VI. e


3. A good way to remember something is to explain it to someone else. Suppose that you arestudying with a classmate who is puzzled when asked to evaluate ln e3. How would youexplain to him an easy way to figure this out? Sample answer: ln means naturallog. The natural log of e3 is the power to which you raise e to get e3. Thisis obviously 3.

1"e

ENTER) LN


Approximations for ! and eThe following expression can be used to approximate e. If greater and greatervalues of n are used, the value of the expression approximates e more andmore closely.

!1 ! "n1

""n

Another way to approximate e is to use this infinite sum. The greater thevalue of n, the closer the approximation.

e # 1 ! 1 ! "12" ! "2

1$ 3" ! "2 $

13 $ 4" ! … ! "2 $ 3 $ 4

1$ … $ n" ! …

In a similar manner, % can be approximated using an infinite productdiscovered by the English mathematician John Wallis (1616–1703).

"%2" # "

21" $ "

23" $ "

43" $ "

45" $ "

65" $ "

67" $ … $ "2n

2&n

1" $ "2n2!n

1" …

Solve each problem.

1. Use a calculator with an ex key to find e to 7 decimal places.

2. Use the expression !1 ! "n1

""nto approximate e to 3 decimal places. Use

5, 100, 500, and 7000 as values of n.

3. Use the infinite sum to approximate e to 3 decimal places. Use the whole numbers from 3 through 6 as values of n.

4. Which approximation method approaches the value of e more quickly?

5. Use a calculator with a % key to find % to 7 decimal places.

6. Use the infinite product to approximate % to 3 decimal places. Use the whole numbers from 3 through 6 as values of n.

7. Does the infinite product give good approximations for % quickly?

8. Show that % 4 ! % 5 is equal to e6 to 4 decimal places.

9. Which is larger, e% or % e?

10. The expression x reaches a maximum value at x # e. Use this fact to prove the inequality you found in Exercise 9.

Enrichment

NAME ______________________________________________ DATE______________ PERIOD _____

10-510-5

Study Guide and InterventionExponential Growth and Decay

NAME ______________________________________________ DATE______________ PERIOD _____

10-610-6


Less

on

10-

6Exponential Decay Depreciation of value and radioactive decay are examples ofexponential decay. When a quantity decreases by a fixed percent each time period, theamount of the quantity after t time periods is given by y # a(1 & r)t, where a is the initialamount and r is the percent decrease expressed as a decimal.Another exponential decay model often used by scientists is y # ae&kt, where k is a constant.

CONSUMER PRICES As technology advances, the price of manytechnological devices such as scientific calculators and camcorders goes down.One brand of hand-held organizer sells for $89.

a. If its price decreases by 6% per year, how much will it cost after 5 years?Use the exponential decay model with initial amount $89, percent decrease 0.06, andtime 5 years.y # a(1 & r)t Exponential decay formula

y # 89(1 & 0.06)5 a # 89, r # 0.06, t # 5

y # $65.32After 5 years the price will be $65.32.

b. After how many years will its price be $50?To find when the price will be $50, again use the exponential decay formula and solve for t.

y # a(1 & r)t Exponential decay formula

50 # 89(1 & 0.06)t y # 50, a # 89, r # 0.06

# (0.94)t Divide each side by 89.

log ! " # log (0.94)t Property of Equality for Logarithms

log ! " # t log 0.94 Power Property

t # Divide each side by log 0.94.

t # 9.3The price will be $50 after about 9.3 years.

1. BUSINESS A furniture store is closing out its business. Each week the owner lowersprices by 25%. After how many weeks will the sale price of a $500 item drop below $100?6 weeks

CARBON DATING Use the formula y ! ae"0.00012t, where a is the initial amount ofCarbon-14, t is the number of years ago the animal lived, and y is the remainingamount after t years.

2. How old is a fossil remain that has lost 95% of its Carbon-14? about 25,000 years old

3. How old is a skeleton that has 95% of its Carbon-14 remaining? about 427.5 years old

log !"5809""

""log 0.94

50"89

50"89

50"89

ExampleExample

ExercisesExercises


Exponential Growth Population increase and growth of bacteria colonies are examplesof exponential growth. When a quantity increases by a fixed percent each time period, theamount of that quantity after t time periods is given by y # a(1 ! r)t, where a is the initialamount and r is the percent increase (or rate of growth) expressed as a decimal.Another exponential growth model often used by scientists is y # aekt, where k is a constant.

A computer engineer is hired for a salary of $28,000. If she gets a5% raise each year, after how many years will she be making $50,000 or more?Use the exponential growth model with a # 28,000, y # 50,000, and r # 0.05 and solve for t.

y # a(1 ! r)t Exponential growth formula

50,000 # 28,000(1 ! 0.05)t y # 50,000, a # 28,000, r # 0.05

# (1.05)t Divide each side by 28,000.

log ! " # log (1.05)t Property of Equality of Logarithms

log ! " # t log 1.05 Power Property

t # Divide each side by log 1.05.

t # 11.9 years Use a calculator.

If raises are given annually, she will be making over $50,000 in 12 years.

1. BACTERIA GROWTH A certain strain of bacteria grows from 40 to 326 in 120 minutes.Find k for the growth formula y # aekt, where t is in minutes. about 0.0175

2. INVESTMENT Carl plans to invest $500 at 8.25% interest, compounded continuously.How long will it take for his money to triple? about 14 years

3. SCHOOL POPULATION There are currently 850 students at the high school, whichrepresents full capacity. The town plans an addition to house 400 more students. If the school population grows at 7.8% per year, in how many years will the new additionbe full? about 5 years

4. EXERCISE Hugo begins a walking program by walking mile per day for one week.

Each week thereafter he increases his mileage by 10%. After how many weeks is hewalking more than 5 miles per day? 24 weeks

5. VOCABULARY GROWTH When Emily was 18 months old, she had a 10-wordvocabulary. By the time she was 5 years old (60 months), her vocabulary was 2500 words.If her vocabulary increased at a constant percent per month, what was that increase?about 14%

1"2

log !"5208""

"log 1.05

50"28

50"28

50"28


Exponential Growth and Decay

NAME ______________________________________________ DATE______________ PERIOD _____

10-610-6

ExampleExample

ExercisesExercises

Skills PracticeExponential Growth and Decay

NAME ______________________________________________ DATE______________ PERIOD _____

10-610-6


Less

on

10-

6Solve each problem.

1. FISHING In an over-fished area, the catch of a certain fish is decreasing at an averagerate of 8% per year. If this decline persists, how long will it take for the catch to reachhalf of the amount before the decline? about 8.3 yr

2. INVESTING Alex invests $2000 in an account that has a 6% annual rate of growth. Tothe nearest year, when will the investment be worth $3600? 10 yr

3. POPULATION A current census shows that the population of a city is 3.5 million. Usingthe formula P # aert, find the expected population of the city in 30 years if the growthrate r of the population is 1.5% per year, a represents the current population in millions,and t represents the time in years. about 5.5 million

4. POPULATION The population P in thousands of a city can be modeled by the equationP # 80e0.015t, where t is the time in years. In how many years will the population of thecity be 120,000? about 27 yr

5. BACTERIA How many days will it take a culture of bacteria to increase from 2000 to50,000 if the growth rate per day is 93.2%? about 4.9 days

6. NUCLEAR POWER The element plutonium-239 is highly radioactive. Nuclear reactorscan produce and also use this element. The heat that plutonium-239 emits has helped topower equipment on the moon. If the half-life of plutonium-239 is 24,360 years, what isthe value of k for this element? about 0.00002845

7. DEPRECIATION A Global Positioning Satellite (GPS) system uses satellite informationto locate ground position. Abu’s surveying firm bought a GPS system for $12,500. TheGPS depreciated by a fixed rate of 6% and is now worth $8600. How long ago did Abubuy the GPS system? about 6.0 yr

8. BIOLOGY In a laboratory, an organism grows from 100 to 250 in 8 hours. What is thehourly growth rate in the growth formula y # a(1 ! r) t? about 12.13%


Solve each problem.

1. INVESTING The formula A # P!1 ! "2tgives the value of an investment after t years in

an account that earns an annual interest rate r compounded twice a year. Suppose $500is invested at 6% annual interest compounded twice a year. In how many years will theinvestment be worth $1000? about 11.7 yr

2. BACTERIA How many hours will it take a culture of bacteria to increase from 20 to2000 if the growth rate per hour is 85%? about 7.5 h

3. RADIOACTIVE DECAY A radioactive substance has a half-life of 32 years. Find theconstant k in the decay formula for the substance. about 0.02166

4. DEPRECIATION A piece of machinery valued at $250,000 depreciates at a fixed rate of12% per year. After how many years will the value have depreciated to $100,000?about 7.2 yr

5. INFLATION For Dave to buy a new car comparably equipped to the one he bought 8 yearsago would cost $12,500. Since Dave bought the car, the inflation rate for cars like his hasbeen at an average annual rate of 5.1%. If Dave originally paid $8400 for the car, howlong ago did he buy it? about 8 yr

6. RADIOACTIVE DECAY Cobalt, an element used to make alloys, has several isotopes.One of these, cobalt-60, is radioactive and has a half-life of 5.7 years. Cobalt-60 is used totrace the path of nonradioactive substances in a system. What is the value of k forCobalt-60? about 0.1216

7. WHALES Modern whales appeared 5&10 million years ago. The vertebrae of a whalediscovered by paleontologists contain roughly 0.25% as much carbon-14 as they wouldhave contained when the whale was alive. How long ago did the whale die? Use k # 0.00012. about 50,000 yr

8. POPULATION The population of rabbits in an area is modeled by the growth equationP(t) # 8e0.26t, where P is in thousands and t is in years. How long will it take for thepopulation to reach 25,000? about 4.4 yr

9. DEPRECIATION A computer system depreciates at an average rate of 4% per month. Ifthe value of the computer system was originally $12,000, in how many months is itworth $7350? about 12 mo

10. BIOLOGY In a laboratory, a culture increases from 30 to 195 organisms in 5 hours.What is the hourly growth rate in the growth formula y # a(1 ! r) t? about 45.4%

r"2

Practice (Average)

Exponential Growth and Decay

NAME ______________________________________________ DATE______________ PERIOD _____

10-610-6

Reading to Learn MathematicsExponential Growth and Decay

NAME ______________________________________________ DATE______________ PERIOD _____

10-610-6


Less

on

10-

6Pre-Activity How can you determine the current value of your car?


• Between which two years shown in the table did the car depreciate bythe greatest amount?between years 0 and 1

• Describe two ways to calculate the value of the car 6 years after it waspurchased. (Do not actually calculate the value.)Sample answer: 1. Multiply $9200.66 by 0.16 and subtract theresult from $9200.66. 2. Multiply $9200.66 by 0.84.

Reading the Lesson

1. State whether each situation is an example of exponential growth or decay.

a. A city had 42,000 residents in 1980 and 128,000 residents in 2000. growth

b. Raul compared the value of his car when he bought it new to the value when hetraded ‘;lpit in six years later. decay

c. A paleontologist compared the amount of carbon-14 in the skeleton of an animalwhen it died to the amount 300 years later. decay

d. Maria deposited $750 in a savings account paying 4.5% annual interest compoundedquarterly. She did not make any withdrawals or further deposits. She compared thebalance in her passbook immediately after she opened the account to the balance 3 years later. growth

2. State whether each equation represents exponential growth or decay.

a. y # 5e0.15t growth b. y # 1000(1 & 0.05) t decay

c. y # 0.3e&1200t decay d. y # 2(1 ! 0.0001) t growth


3. Visualizing their graphs is often a good way to remember the difference betweenmathematical equations. How can your knowledge of the graphs of exponential equationsfrom Lesson 10-1 help you to remember that equations of the form y # a(1 ! r) t

represent exponential growth, while equations of the form y # a(1 & r) t representexponential decay?Sample answer: If a # 0, the graph of y ! abx is always increasing if b # 1 and is always decreasing if 0 $ b $ 1. Since r is always a positivenumber, if b ! 1 % r, the base will be greater than 1 and the function willbe increasing (growth), while if b ! 1 " r, the base will be less than 1and the function will be decreasing (decay).


Effective Annual YieldWhen interest is compounded more than once per year, the effective annualyield is higher than the annual interest rate. The effective annual yield, E, isthe interest rate that would give the same amount of interest if the interestwere compounded once per year. If P dollars are invested for one year, thevalue of the investment at the end of the year is A # P(1 ! E). If P dollarsare invested for one year at a nominal rate r compounded n times per year,

the value of the investment at the end of the year is A # P!1 ! "nr

""n. Setting

the amounts equal and solving for E will produce a formula for the effectiveannual yield.

P(1 ! E) # P!1 ! "nr

""n

1 ! E # !1 ! "nr

""n

E # !1 ! "nr

""n& 1

If compounding is continuous, the value of the investment at the end of oneyear is A # Per. Again set the amounts equal and solve for E. A formula forthe effective annual yield under continuous compounding is obtained.

P(1 ! E) # Per

1 ! E # er

E # er & 1

Enrichment

NAME ______________________________________________ DATE______________ PERIOD _____

10-610-6

Find the effectiveannual yield of an investment made at7.5% compounded monthly.r # 0.075n # 12

E # !1 ! "0.

10275""12

& 1 $ 7.76%

Find the effectiveannual yield of an investment made at6.25% compounded continuously.r # 0.0625

E # e0.0625 & 1 $ 6.45%


Find the effective annual yield for each investment.

1. 10% compounded quarterly 2. 8.5% compounded monthly

3. 9.25% compounded continuously 4. 7.75% compounded continuously

5. 6.5% compounded daily (assume a 365-day year)

6. Which investment yields more interest—9% compounded continuously or 9.2% compounded quarterly?

© Glencoe/McGraw-Hill A2 Glencoe Algebra 2

Answers (Lesson 10-1)

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ality

for

If b

is a

pos

itive

num

ber

othe

r th

an 1

,E

xpon

entia

l Fun

ctio

nsth

en b

x!

by

if an

d on

ly if

x!

y.

Pro

pert

y of

Ineq

ualit

y fo

rIf

b#

1

Exp

onen

tial F

unct

ions

then

bx

#b

yif

and

only

if x

#y

and

bx$

by

if an

d on

ly if

x$

y.

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Exp

onen

tial F

unct

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-1

10-1

Sol

ve 4

x"

1!

2x#

5 .4x

%1

!2x

(5

Orig

inal

equ

atio

n

(22 )

x%

1!

2x(

5R

ewrit

e 4

as 2

2 .

2(x

%1)

!x

(5

Pro

p. o

f Ine

qual

ity fo

r E

xpon

entia

lF

unct

ions

2x%

2 !

x(

5D

istr

ibut

ive

Pro

pert

y

x!

7S

ubtr

act x

and

add

2 to

eac

h si

de.

Sol

ve 5

2x"

1$

.

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%1

#O

rigin

al in

equa

lity

52x

%1

#5%

3R

ewrit

e as

5%

3 .

2x%

1 #

%3

Pro

p. o

f Ine

qual

ity fo

r E

xpon

entia

l Fun

ctio

ns

2x#

%2

Add

1 to

eac

h si

de.

x#

%1

Div

ide

each

sid

e by

2.

The

sol

utio

n se

t is

{x|x

#%

1}.

1& 12

5

1& 12

5

1% 12

5Ex

ampl

e1Ex

ampl

e1Ex

ampl

e2Ex

ampl

e2

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cises

Exer

cises

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pli

fy e

ach

exp

ress

ion

.

1.(3#

2$ )#2$

2.25

#2$

'12

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2$ y3#

2$ )#2$

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!2"

or 3

125!

2"x2

y6

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6$ )(x#

5$ )5.

(x#6$ )#

5$6.

(2x*

)(5x3

*)

x!6"

#!

5"x!

30"10

x4&

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ve e

ach

equ

atio

n o

r in

equ

alit

y.C

hec

k y

our

solu

tion

.

7.32

x%

1!

3x(

23

8.23

x!

4x(

24

9.32

x%

1!

"

10.4

x(

1!

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(3

"11

.8x

%2

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.252

x!

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6

13.4

#x$

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2014

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615

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2$!

81& #1 8$&

3

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x$

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2x$

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%5

x$

15

19.1

04x

(1

#10

0x%

220

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x221

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%5

$4x

(8

x$

"x

$or

x'

0x

'%3 41 %

3 % 25 % 2

5 % 31 & 27

2 % 31 & 16

7 % 4

1 % 21 & 9


A


Skil

ls P

ract

ice

Exp

onen

tial F

unct

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-1

10-1

©G

lenc

oe/M

cGra

w-H

ill57

5G

lenc

oe A

lgeb

ra 2

Lesson 10-1

Sk

etch

th

e gr

aph

of

each

fu

nct

ion

.Th

en s

tate

th

e fu

nct

ion

’s d

omai

n a

nd

ran

ge.

1.y

!3(

2)x

2.y

!2 !

"x

dom

ain:

all r

eal n

umbe

rs;

dom

ain:

all r

eal n

umbe

rs;

rang

e:al

l pos

itive

num

bers

rang

e:al

l pos

itive

num

bers

Det

erm

ine

wh

eth

er e

ach

fu

nct

ion

rep

rese

nts

exp

onen

tial

gro

wth

or d

eca

y.

3.y

!3(

6)x

grow

th4.

y!

2 !"x

deca

y

5.y

!10

%x

deca

y6.

y!

2(2.

5)x

grow

th

Wri

te a

n e

xpon

enti

al f

un

ctio

n w

hos

e gr

aph

pas

ses

thro

ugh

th

e gi

ven

poi

nts

.

7.(0

,1)

and

(%1,

3)y

!#

$x8.

(0,4

) an

d (1

,12)

y!

4(3)

x

9.(0

,3)

and

(%1,

6)y

!3 #

$x10

.(0,

5) a

nd (

1,15

)y

!5(

3)x

11.(

0,0.

1) a

nd (

1,0.

5)y

!0.

1(5)

x12

.(0,

0.2)

and

(1,

1.6)

y!

0.2(

8)x

Sim

pli

fy e

ach

exp

ress

ion

.

13. (3

#3$ )#

3$27

14.(x

#2$ )#

7$x!

14"

15.5

2#3$

'54

#3$

56!

3"16

.x3*

)x*

x2&

Sol

ve e

ach

equ

atio

n o

r in

equ

alit

y.C

hec

k y

our

solu

tion

.

17.3

x#

9x

$2

18.2

2x(

3!

321

19.4

9x+

x(

"20

.43x

%2

!16

21.3

2x(

5!

27x

522

.27x

!32

x(

334 % 3

1 % 21 & 7

1 % 21 % 3

9 & 10

x

y

Ox

y

O

1 & 2

©G

lenc

oe/M

cGra

w-H

ill57

6G

lenc

oe A

lgeb

ra 2

Sk

etch

th

e gr

aph

of

each

fu

nct

ion

.Th

en s

tate

th

e fu

nct

ion

’s d

omai

n a

nd

ran

ge.

1.y

!1.

5(2)

x2.

y!

4(3)

x3.

y!

3(0.

5)x

dom

ain:

all r

eal

dom

ain:

all r

eal

dom

ain:

all r

eal

num

bers

;ran

ge:a

ll nu

mbe

rs;r

ange

:all

num

bers

;ran

ge:a

ll po

sitiv

e nu

mbe

rspo

sitiv

e nu

mbe

rspo

sitiv

e nu

mbe

rs

Det

erm

ine

wh

eth

er e

ach

fu

nct

ion

rep

rese

nts

exp

onen

tial

gro

wth

or d

eca

y.

4.y

!5(

0.6)

xde

cay

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1(2)

xgr

owth

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!5

'4%

xde

cay

Wri

te a

n e

xpon

enti

al f

un

ctio

n w

hos

e gr

aph

pas

ses

thro

ugh

th

e gi

ven

poi

nts

.

7.(0

,1)

and

(%1,

4)8.

(0,2

) an

d (1

,10)

9.(0

,%3)

and

(1,

%1.

5)

y!

#$x

y!

2(5)

xy

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3(0.

5)x

10.(

0,0.

8) a

nd (

1,1.

6)11

.(0,

%0.

4) a

nd (

2,%

10)

12.(

0,*

) an

d (3

,8*

)

y!

0.8(

2)x

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4(5)

xy

!&

(2)x

Sim

pli

fy e

ach

exp

ress

ion

.

13.(2

#2$ )#

8$16

14.(n

#3$ )#

75$n1

515

.y#

6$'

y5#

6$y

6!6"

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3#6$

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#24$

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6"17

.n3

)n*

n3

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25#

11$)

5#11$

52!

11"

Sol

ve e

ach

equ

atio

n o

r in

equ

alit

y.C

hec

k y

our

solu

tion

.

19.3

3x %

5#

81x

$3

20.7

6x!

72x

%20

"5

21.3

6n%

5$

94n

%3

n$

22.9

2x%

1!

27x

(4

1423

.23n

%1

,!

"nn

)24

.164

n%

1!

1282

n(

1

BIO

LOG

YF

or E

xerc

ises

25

and

26,

use

th

e fo

llow

ing

info

rmat

ion

.T

he in

itia

l num

ber

of b

acte

ria

in a

cul

ture

is 1

2,00

0.T

he n

umbe

r af

ter

3 da

ys is

96,

000.

25.W

rite

an

expo

nent

ial f

unct

ion

to m

odel

the

pop

ulat

ion

yof

bac

teri

a af

ter

xda

ys.

y!

12,0

00(2

)x26

.How

man

y ba

cter

ia a

re t

here

aft

er 6

day

s?76

8,00

027

.ED

UC

ATI

ON

A c

olle

ge w

ith

a gr

adua

ting

cla

ss o

f 40

00 s

tude

nts

in t

he y

ear

2002

pred

icts

tha

t it

will

hav

e a

grad

uati

ng c

lass

of 4

862

in 4

yea

rs.W

rite

an

expo

nent

ial

func

tion

to

mod

el t

he n

umbe

r of

stu

dent

s y

in t

he g

radu

atin

g cl

ass

tye

ars

afte

r 20

02.

y!

4000

(1.0

5)t

11 % 21 % 6

1 & 8

1 % 2

1 % 4

x

y

Ox

y

Ox

y

OPra

ctic

e (A

vera

ge)

Exp

onen

tial F

unct

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-1

10-1



Rea

din

g t

o L

earn

Math

emati

csE

xpon

entia

l Fun

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-1

10-1

©G

lenc

oe/M

cGra

w-H

ill57

7G

lenc

oe A

lgeb

ra 2

Lesson 10-1

Pre-

Act

ivit

yH

ow d

oes

an e

xpon

enti

al f

un

ctio

n d

escr

ibe

tou

rnam

ent

pla

y?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 10

-1 a

t th

e to

p of

pag

e 52

3 in

you

r te

xtbo

ok.

How

man

y ro

unds

of

play

wou

ld b

e ne

eded

for

a t

ourn

amen

t w

ith

100

play

ers?

7

Rea

din

g t

he

Less

on

1.In

dica

te w

heth

er e

ach

of t

he f

ollo

win

g st

atem

ents

abo

ut t

he e

xpon

enti

al f

unct

ion

y!

10x

is t

rue

or f

alse

.

a.T

he d

omai

n is

the

set

of

all p

osit

ive

real

num

bers

.fa

lse

b.T

he y

-int

erce

pt is

1.

true

c.T

he f

unct

ion

is o

ne-t

o-on

e.tr

ue

d.T

he y

-axi

s is

an

asym

ptot

e of

the

gra

ph.

fals

e

e.T

he r

ange

is t

he s

et o

f al

l rea

l num

bers

.fa

lse

2.D

eter

min

e w

heth

er e

ach

func

tion

rep

rese

nts

expo

nent

ial g

row

thor

dec

ay.

a.y

!0.

2(3)

x .gr

owth

b.y

!3 !

"x .de

cay

c.y

!0.

4(1.

01)x

.gr

owth

3.Su

pply

the

rea

son

for

each

ste

p in

the

fol

low

ing

solu

tion

of

an e

xpon

enti

al e

quat

ion.

92x

%1

!27

xO

rigi

nal e

quat

ion

(32 )

2x%

1!

(33 )

xR

ewri

te e

ach

side

with

a b

ase

of 3

.32

(2x

%1)

!33

xP

ower

of

a P

ower

2(2x

%1)

!3x

Pro

pert

y of

Equ

ality

for

Exp

onen

tial F

unct

ions

4x%

2 !

3xD

istr

ibut

ive

Pro

pert

yx

%2

!0

Sub

trac

t 3x

from

eac

h si

de.

x!

2A

dd 2

to

each

sid

e.

Hel

pin

g Y

ou

Rem

emb

er

4.O

ne w

ay t

o re

mem

ber

that

pol

ynom

ial f

unct

ions

and

exp

onen

tial

fun

ctio

ns a

re d

iffe

rent

is t

o co

ntra

st t

he p

olyn

omia

l fun

ctio

n y

!x2

and

the

expo

nent

ial f

unct

ion

y!

2x.T

ell a

tle

ast

thre

e w

ays

they

are

dif

fere

nt.

Sam

ple

answ

er:I

n y

!x2

,the

var

iabl

e x

is a

bas

e,bu

t in

y!

2x,t

heva

riab

le x

is a

n ex

pone

nt.T

he g

raph

of

y!

x2is

sym

met

ric

with

res

pect

to t

he y

-axi

s,bu

t th

e gr

aph

of y

!2x

is n

ot.T

he g

raph

of y

!x2

touc

hes

the

x-ax

is a

t (0,

0),b

ut th

e gr

aph

of y

!2x

has

the

x-ax

is a

s an

asy

mpt

ote.

You

can

com

pute

the

val

ue o

f y

!x2

men

tally

for

x!

100,

but

you

cann

otco

mpu

te t

he v

alue

of

y!

2xm

enta

lly fo

r x

!10

0.

2 & 5

©G

lenc

oe/M

cGra

w-H

ill57

8G

lenc

oe A

lgeb

ra 2

Find

ing

Sol

utio

ns o

f xy

!yx

Perh

aps

you

have

not

iced

tha

t if

xan

d y

are

inte

rcha

nged

in e

quat

ions

suc

has

x!

yan

d xy

!1,

the

resu

ltin

g eq

uati

on is

equ

ival

ent

to t

he o

rigi

nal

equa

tion

.The

sam

e is

tru

e of

the

equ

atio

n xy

!yx

.How

ever

,fin

ding

solu

tion

s of

xy

!yx

and

draw

ing

its

grap

h is

not

a s

impl

e pr

oces

s.

Sol

ve e

ach

pro

blem

.Ass

um

e th

at x

and

yar

e p

osit

ive

real

nu

mbe

rs.

1.If

a#

0,w

ill (

a,a)

be

a so

luti

on o

f xy

!yx

? Ju

stif

y yo

ur a

nsw

er.

Yes,

sinc

e aa

!aa

mus

t be

tru

e (R

efle

xive

Pro

p.of

Equ

ality

).

2.If

c#

0,d

#0,

and

(c,d

) is

a s

olut

ion

of x

y!

yx,w

ill (

d,c)

als

o be

a s

olut

ion?

Jus

tify

you

r an

swer

.

Yes;

repl

acin

g x

with

d,y

with

cgi

ves

dc

!cd

;but

if (

c,d)

is a

sol

utio

n,cd

!d

c .S

o,by

the

Sym

met

ric

Pro

pert

y of

Equ

ality

,dc

!cd

is t

rue.

3.U

se 2

as

a va

lue

for

yin

xy

!yx

.The

equ

atio

n be

com

es x

2!

2x.

a.F

ind

equa

tion

s fo

r tw

o fu

ncti

ons,

f(x)

and

g(x

) th

at y

ou c

ould

gra

ph t

ofi

nd t

he s

olut

ions

of x

2!

2x.T

hen

grap

h th

e fu

ncti

ons

on a

sep

arat

esh

eet

of g

raph

pap

er.

f(x)

!x2

,g(x

) !

2xS

ee s

tude

nts’

grap

hs.

b.U

se t

he g

raph

you

dre

w f

or p

art

ato

sta

te t

wo

solu

tion

s fo

r x2

!2x

.T

hen

use

thes

e so

luti

ons

to s

tate

tw

o so

luti

ons

for

xy!

yx.

2,4;

(2,2

),(4

,2)

4.In

thi

s ex

erci

se,a

gra

phin

g ca

lcul

ator

will

be

very

hel

pful

.Use

the

te

chni

que

of E

xerc

ise

3 to

com

plet

e th

e ta

bles

bel

ow.T

hen

grap

h xy

!yx

for

posi

tive

val

ues

of x

and

y.If

the

re a

re a

sym

ptot

es,s

how

the

m in

you

rdi

agra

m u

sing

dot

ted

lines

.Not

e th

at in

the

tab

le,s

ome

valu

es o

f yca

llfo

r on

e va

lue

of x

,oth

ers

call

for

two.

x

y

O

xy

44

24

55

1.8

5

88

1.5

8

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-1

10-1 x

y

%1 2%&1 2&

%3 4%&3 4&

11

22

42

33

2.5

3


A


Stu

dy

Gu

ide

an

d I

nte

rven

tion

Loga

rith

ms

and

Loga

rith

mic

Fun

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-2

10-2

©G

lenc

oe/M

cGra

w-H

ill57

9G

lenc

oe A

lgeb

ra 2

Lesson 10-2

Log

arit

hm

ic F

un

ctio

ns

and

Exp

ress

ion

s

Def

initi

on o

f Lo

gari

thm

Le

t ban

d x

be p

ositi

ve n

umbe

rs, b

"1.

The

loga

rithm

of x

with

bas

e b

is d

enot

ed

with

Bas

e b

log b

xan

d is

def

ined

as

the

expo

nent

yth

at m

akes

the

equa

tion

by!

xtr

ue.

The

inve

rse

of t

he e

xpon

enti

al f

unct

ion

y!

bxis

the

log

arit

hm

ic f

un

ctio

nx

!by

.T

his

func

tion

is u

sual

ly w

ritt

en a

s y

!lo

g bx.

1.T

he fu

nctio

n is

con

tinuo

us a

nd o

ne-t

o-on

e.

Pro

pert

ies

of2.

The

dom

ain

is th

e se

t of a

ll po

sitiv

e re

al n

umbe

rs.

Loga

rith

mic

Fun

ctio

ns3.

The

y-a

xis

is a

n as

ympt

ote

of th

e gr

aph.

4.T

he r

ange

is th

e se

t of a

ll re

al n

umbe

rs.

5.T

he g

raph

con

tain

s th

e po

int (

0, 1

).

Wri

te a

n e

xpon

enti

al e

quat

ion

equ

ival

ent

to l

og3

243

!5.

35!

243

Wri

te a

log

arit

hm

ic e

quat

ion

equ

ival

ent

to 6

"3

!.

log 6

!%

3

Eva

luat

e lo

g 816

.

8&4 3&

!16

,so

log 8

16 !

.

Wri

te e

ach

equ

atio

n i

n l

ogar

ith

mic

for

m.

1.27

!12

82.

3%4

!3.

!"3

!

log 2

128

!7

log 3

!"

4lo

g%1 7%

!3

Wri

te e

ach

equ

atio

n i

n e

xpon

enti

al f

orm

.

4.lo

g 15

225

!2

5.lo

g 3!

%3

6.lo

g 432

!

152

!22

53"

3!

4%5 2%!

32

Eva

luat

e ea

ch e

xpre

ssio

n.

7.lo

g 464

38.

log 2

646

9.lo

g 100

100,

000

2.5

10.l

og5

625

411

.log

2781

12.l

og25

5

13.l

og2

"7

14.l

og10

0.00

001

"5

15.l

og4

"2.

51 & 32

1& 12

8

1 % 24 % 31 % 27

5 & 21 & 27

1% 34

31 % 81

1& 34

31 & 7

1 & 81

4 & 3

1& 21

6

1% 21

6

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exam

ple3

Exam

ple3

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill58

0G

lenc

oe A

lgeb

ra 2

Solv

e Lo

gar

ith

mic

Eq

uat

ion

s an

d In

equ

alit

ies

Loga

rith

mic

to

If b

#1,

x#

0, a

nd lo

g bx

#y,

then

x#

by.

Exp

onen

tial I

nequ

ality

If b

#1,

x#

0, a

nd lo

g bx

$y,

then

0 $

x$

by.

Pro

pert

y of

Equ

ality

for

If b

is a

pos

itive

num

ber

othe

r th

an 1

, Lo

gari

thm

ic F

unct

ions

then

log b

x!

log b

yif

and

only

if x

!y.

Pro

pert

y of

Ineq

ualit

y fo

r If

b#

1, th

en lo

g bx

#lo

g by

if an

d on

ly if

x#

y,

Loga

rith

mic

Fun

ctio

nsan

d lo

g bx

$lo

g by

if an

d on

ly if

x$

y.

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Loga

rith

ms

and

Loga

rith

mic

Fun

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-2

10-2

Sol

ve l

og2

2x!

3.lo

g 22x

!3

Orig

inal

equ

atio

n

2x!

23D

efin

ition

of l

ogar

ithm

2x!

8S

impl

ify.

x!

4S

impl

ify.

The

sol

utio

n is

x!

4.

Sol

ve l

og5

(4x

"3)

'3.

log 5

(4x

%3)

$3

Orig

inal

equ

atio

n

0 $

4x%

3 $

53Lo

garit

hmic

to e

xpon

entia

l ine

qual

ity

3 $

4x$

125

(3

Add

ition

Pro

pert

y of

Ineq

ualit

ies

$x

$32

Sim

plify

.

The

sol

utio

n se

t is

'x|$

x$

32(.

3 & 4

3 & 4

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sol

ve e

ach

equ

atio

n o

r in

equ

alit

y.

1.lo

g 232

!3x

2.lo

g 32c

!%

2

3.lo

g 2x

16 !

%2

4.lo

g 25!

"!10

5.lo

g 4(5

x(

1) !

23

6.lo

g 8(x

%5)

!9

7.lo

g 4(3

x%

1) !

log 4

(2x

(3)

48.

log 2

(x2

%6)

!lo

g 2(2

x(

2)4

9.lo

g x(

427

!3

"1

10.l

og2

(x(

3) !

413

11.l

ogx

1000

!3

1012

.log

8(4

x(

4) !

215

13.l

og2

2x#

2x

$2

14.l

og5

x#

2x

$25

15.l

og2

(3x

(1)

$4

"'

x'

516

.log

4(2

x) #

%x

$

17.l

og3

(x(

3) $

3"

3 '

x'

2418

.log

276x

#x

$3 % 2

2 & 3

1 % 41 & 2

1 % 3

2 & 3

1 & 2x & 2

1 % 8

1 % 185 % 3



Skil

ls P

ract

ice

Loga

rith

ms

and

Loga

rith

mic

Fun

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-2

10-2

©G

lenc

oe/M

cGra

w-H

ill58

1G

lenc

oe A

lgeb

ra 2

Lesson 10-2

Wri

te e

ach

equ

atio

n i

n l

ogar

ith

mic

for

m.

1.23

!8

log 2

8 !

32.

32!

9lo

g 39

!2

3.8%

2!

log 8

!"

24.

!"2

!lo

g %1 3%

!2

Wri

te e

ach

equ

atio

n i

n e

xpon

enti

al f

orm

.

5.lo

g 324

3 !

535

!24

36.

log 4

64 !

343

!64

7.lo

g 93

!9%1 2%

!3

8.lo

g 5!

%2

5"2

!

Eva

luat

e ea

ch e

xpre

ssio

n.

9.lo

g 525

210

.log

93

11.l

og10

1000

312

.log

125

5

13.l

og4

"3

14.l

og5

"4

15.l

og8

833

16.l

og27

"

Sol

ve e

ach

equ

atio

n o

r in

equ

alit

y.C

hec

k y

our

solu

tion

s.

17.l

og3

x!

524

318

.log

2x

!3

8

19.l

og4

y$

00

'y

'1

20.l

og&1 4&

x!

3

21.l

og2

n#

%2

n$

22.l

ogb

3 !

9

23.l

og6

(4x

(12

) !

26

24.l

og2

(4x

%4)

#5

x$

9

25.l

og3

(x(

2) !

log 3

(3x)

126

.log

6(3

y%

5) ,

log 6

(2y

(3)

y)

8

1 & 21 % 4

1 % 641 % 31 & 31

& 625

1 & 64

1 % 3

1 % 2

1 % 251 & 25

1 & 2

1 % 91 & 9

1 & 31 % 64

1 & 64

©G

lenc

oe/M

cGra

w-H

ill58

2G

lenc

oe A

lgeb

ra 2

Wri

te e

ach

equ

atio

n i

n l

ogar

ith

mic

for

m.

1.53

!12

5lo

g 512

5 !

32.

70!

1lo

g 71

!0

3.34

!81

log 3

81 !

4

4.3%

4!

5.!

"3!

6.77

76&1 5&

!6

log 3

!"

4lo

g %1 4%

!3

log 7

776

6 !

Wri

te e

ach

equ

atio

n i

n e

xpon

enti

al f

orm

.

7.lo

g 621

6 !

363

!21

68.

log 2

64 !

626

!64

9.lo

g 3!

%4

3"4

!

10.l

og10

0.00

001

!%

511

.log

255

!12

.log

328

!

10"

5!

0.00

001

25%1 2%

!5

32%3 5%

!8

Eva

luat

e ea

ch e

xpre

ssio

n.

13.l

og3

814

14.l

og10

0.00

01"

415

.log

2"

416

.log

&1 3&27

"3

17.l

og9

10

18.l

og8

419

.log

7"

220

.log

664

4

21.l

og3

"1

22.l

og4

"4

23.l

og9

9(n

(1)

n#

124

.2lo

g 232

32

Sol

ve e

ach

equ

atio

n o

r in

equ

alit

y.C

hec

k y

our

solu

tion

s.

25.l

og10

n!

%3

26.l

og4

x#

3x

$64

27.l

og4

x!

8

28.l

og&1 5&

x!

%3

125

29.l

og7

q$

00

'q

'1

30.l

og6

(2y

(8)

,2

y)

14

31.l

ogy

16 !

%4

32.l

ogn

!%

32

33.l

ogb

1024

!5

4

34.l

og8

(3x

(7)

$lo

g 8(7

x(

4)35

.log

7(8

x(

20) !

log 7

(x(

6)36

.log

3(x

2%

2) !

log 3

x

x$

"2

2

37.S

OU

ND

Soun

ds t

hat

reac

h le

vels

of

130

deci

bels

or

mor

e ar

e pa

infu

l to

hum

ans.

Wha

tis

the

rel

ativ

e in

tens

ity

of 1

30 d

ecib

els?

1013

38.I

NV

ESTI

NG

Mar

ia in

vest

s $1

000

in a

sav

ings

acc

ount

tha

t pa

ys 8

% in

tere

stco

mpo

unde

d an

nual

ly.T

he v

alue

of

the

acco

unt

Aat

the

end

of

five

yea

rs c

an b

ede

term

ined

fro

m t

he e

quat

ion

log

A!

log[

1000

(1 (

0.08

)5].

Fin

d th

e va

lue

of A

to t

hene

ares

t do

llar.

$146

9

3 % 4

1 & 81 % 2

3 & 21

% 1000

1& 25

61 & 3

1 & 492 % 3

1 & 16

3 & 51 & 2

1 % 811 & 81

1 % 51 % 64

1 % 81

1 & 641 & 4

1 & 81Pra

ctic

e (A

vera

ge)

Loga

rith

ms

and

Loga

rith

mic

Fun

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-2

10-2


A


Rea

din

g t

o L

earn

Math

emati

csLo

gari

thm

s an

d Lo

gari

thm

ic F

unct

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-2

10-2

©G

lenc

oe/M

cGra

w-H

ill58

3G

lenc

oe A

lgeb

ra 2

Lesson 10-2

Pre-

Act

ivit

yW

hy

is a

log

arit

hm

ic s

cale

use

d t

o m

easu

re s

oun

d?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 10

-2 a

t th

e to

p of

pag

e 53

1 in

you

r te

xtbo

ok.

How

man

y ti

mes

loud

er t

han

a w

hisp

er is

nor

mal

con

vers

atio

n?10

4or

10,

000

times

Rea

din

g t

he

Less

on

1.a.

Wri

te a

n ex

pone

ntia

l equ

atio

n th

at is

equ

ival

ent

to lo

g 381

!4.

34!

81

b.W

rite

a lo

gari

thm

ic e

quat

ion

that

is e

quiv

alen

t to

25%

&1 2&!

.lo

g 25

!"

c.W

rite

an

expo

nent

ial e

quat

ion

that

is e

quiv

alen

t to

log 4

1 !

0.40

!1

d.W

rite

a lo

gari

thm

ic e

quat

ion

that

is e

quiv

alen

t to

10%

3!

0.00

1.lo

g 10

0.00

1 !

"3

e.W

hat

is t

he in

vers

e of

the

fun

ctio

n y

!5x

?y

!lo

g 5x

f.W

hat

is t

he in

vers

e of

the

fun

ctio

n y

!lo

g 10

x?y

!10

x

2.M

atch

eac

h fu

ncti

on w

ith

its

grap

h.

a.y

!3x

IVb.

y!

log 3

xI

c.y

!!

"xII

I.II

.II

I.

3.In

dica

te w

heth

er e

ach

of t

he f

ollo

win

g st

atem

ents

abo

ut t

he e

xpon

enti

al f

unct

ion

y!

log 5

xis

tru

eor

fal

se.

a.T

he y

-axi

s is

an

asym

ptot

e of

the

gra

ph.

true

b.T

he d

omai

n is

the

set

of

all r

eal n

umbe

rs.

fals

ec.

The

gra

ph c

onta

ins

the

poin

t (5

,0).

fals

ed.

The

ran

ge is

the

set

of

all r

eal n

umbe

rs.

true

e.T

he y

-int

erce

pt is

1.

fals

e

Hel

pin

g Y

ou

Rem

emb

er4.

An

impo

rtan

t sk

ill n

eede

d fo

r w

orki

ng w

ith

loga

rith

ms

is c

hang

ing

an e

quat

ion

betw

een

loga

rith

mic

and

exp

onen

tial

form

s.U

sing

the

wor

ds b

ase,

expo

nent

,and

loga

rith

m,d

escr

ibe

an e

asy

way

to

rem

embe

r an

d ap

ply

the

part

of

the

defi

niti

on o

f lo

gari

thm

tha

t sa

ys,

“log

bx

!y

if a

nd o

nly

if b

y!

x.”

Sam

ple

answ

er:I

n th

ese

equa

tions

,bst

ands

for

base

.In

log

form

,bis

the

sub

scri

pt,a

nd in

exp

onen

tial f

orm

,bis

the

num

ber

that

is r

aise

d to

a p

ower

.A lo

gari

thm

is a

n ex

pone

nt,s

o y,

whi

ch is

the

log

in th

e fir

st e

quat

ion,

beco

mes

the

expo

nent

in th

e se

cond

equ

atio

n.

x

y

Ox

y

O

x

y

O

1 & 3

1 % 21 % 5

1 & 5

©G

lenc

oe/M

cGra

w-H

ill58

4G

lenc

oe A

lgeb

ra 2

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-2

10-2

Mus

ical

Rel

atio

nshi

psT

he f

requ

enci

es o

f no

tes

in a

mus

ical

sca

le t

hat

are

one

octa

ve a

part

are

rela

ted

by a

n ex

pone

ntia

l equ

atio

n.Fo

r th

e ei

ght

C n

otes

on

a pi

ano,

the

equa

tion

is C

n!

C12

n%

1 ,w

here

Cn

repr

esen

ts t

he f

requ

ency

of

note

Cn.

1.F

ind

the

rela

tion

ship

bet

wee

n C

1an

d C

2.C

2!

2C1

2.F

ind

the

rela

tion

ship

bet

wee

n C

1an

d C

4.C

4!

8C1

The

fre

quen

cies

of

cons

ecut

ive

note

s ar

e re

late

d by

a

com

mon

rat

io r

.The

gen

eral

equ

atio

n is

f n!

f 1rn

%1 .

3.If

the

fre

quen

cy o

f m

iddl

e C

is 2

61.6

cyc

les

per

seco

nd

and

the

freq

uenc

y of

the

nex

t hi

gher

C is

523

.2 c

ycle

s pe

r se

cond

,fin

d th

e co

mm

on r

atio

r.(

Hin

t:T

he t

wo

C’s

ar

e 12

not

es a

part

.) W

rite

the

ans

wer

as

a ra

dica

lex

pres

sion

.

r!

12 !2"

4.Su

bsti

tute

dec

imal

val

ues

for

ran

d f 1

to f

ind

a sp

ecif

ic

equa

tion

for

f n.

f n!

261.

1(1.

0594

6)n

"1

5.F

ind

the

freq

uenc

y of

F#

abov

e m

iddl

e C

.

f 7!

261.

6(1.

0594

6)6

%36

9.95

6.F

rets

are

a s

erie

s of

rid

ges

plac

ed a

cros

s th

e fi

nger

boar

d of

a g

uita

r.T

hey

are

spac

ed s

o th

at t

he s

ound

mad

e by

pre

ssin

g a

stri

ng a

gain

st o

ne f

ret

has

abou

t 1.

0595

tim

es t

he w

avel

engt

h of

the

sou

nd m

ade

by u

sing

the

next

fre

t.T

he g

ener

al e

quat

ion

is w

n!

w0(

1.05

95)n

.Des

crib

e th

ear

rang

emen

t of

the

fre

ts o

n a

guit

ar.

The

fret

s ar

e sp

aced

in a

lo

gari

thm

ic s

cale

.



Stu

dy

Gu

ide

an

d I

nte

rven

tion

Pro

pert

ies

of L

ogar

ithm

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-3

10-3

©G

lenc

oe/M

cGra

w-H

ill58

5G

lenc

oe A

lgeb

ra 2

Lesson 10-3

Pro

per

ties

of

Log

arit

hm

sP

rope

rtie

s of

exp

onen

ts c

an b

e us

ed t

o de

velo

p th

efo

llow

ing

prop

erti

es o

f lo

gari

thm

s.

Pro

duct

Pro

pert

y F

or a

ll po

sitiv

e nu

mbe

rs m

, n, a

nd b

, whe

re b

"1,

of

Log

arith

ms

log b

mn

!lo

g bm

(lo

g bn.

Quo

tient

Pro

pert

y F

or a

ll po

sitiv

e nu

mbe

rs m

, n, a

nd b

, whe

re b

"1,

of

Log

arith

ms

log b

&m n&!

log b

m%

log b

n.

Pow

er P

rope

rty

For

any

rea

l num

ber

pan

d po

sitiv

e nu

mbe

rs m

and

b,

of L

ogar

ithm

sw

here

b"

1, lo

g bm

p!

plo

g bm

.

Use

log

328

)3.

0331

an

d l

og3

4 )

1.26

19 t

o ap

pro

xim

ate

the

valu

e of

eac

h e

xpre

ssio

n.

Exam

ple

Exam

ple

a.lo

g 336

log 3

36!

log 3

(32

'4)

!lo

g 332

(lo

g 34

!2

(lo

g 34

)2

(1.

2619

)3.

2619

b.lo

g 37

log 3

7!

log 3

!"

!lo

g 328

%lo

g 34

)3.

0331

%1.

2619

)1.

7712

c.lo

g 325

6

log 3

256

!lo

g 3(4

4 )!

4 '

log 3

4)

4(1.

2619

))

5.04

76

28 & 4

Exer

cises

Exer

cises

Use

log

123

)0.

4421

an

d l

og12

7 )

0.78

31 t

o ev

alu

ate

each

exp

ress

ion

.

1.lo

g 12

211.

2252

2.lo

g 12

0.34

103.

log 1

249

1.56

62

4.lo

g 12

361.

4421

5.lo

g 12

631.

6673

6.lo

g 12

"0.

2399

7.lo

g 12

0.20

228.

log 1

216

,807

3.91

559.

log 1

244

12.

4504

Use

log

53

)0.

6826

an

d l

og5

4 )

0.86

14 t

o ev

alu

ate

each

exp

ress

ion

.

10.l

og5

121.

5440

11.l

og5

100

2.86

1412

.log

50.

75"

0.17

88

13.l

og5

144

3.08

8014

.log

50.

3250

15.l

og5

375

3.68

26

16.l

og5

1.3$

0.17

8817

.log

5"

0.35

7618

.log

51.

7304

81 & 59 & 1627 & 16

81 & 49

27 & 49

7 & 3

©G

lenc

oe/M

cGra

w-H

ill58

6G

lenc

oe A

lgeb

ra 2

Solv

e Lo

gar

ith

mic

Eq

uat

ion

sYo

u ca

n us

e th

e pr

oper

ties

of

loga

rith

ms

to s

olve

equa

tion

s in

volv

ing

loga

rith

ms.

Sol

ve e

ach

equ

atio

n.

a.2

log 3

x%

log 3

4 !

log 3

25

2 lo

g 3x

%lo

g 34

!lo

g 325

Orig

inal

equ

atio

n

log 3

x2%

log 3

4 !

log 3

25P

ower

Pro

pert

y

log 3

!lo

g 325

Quo

tient

Pro

pert

y

!25

Pro

pert

y of

Equ

ality

for

Loga

rithm

ic F

unct

ions

x2!

100

Mul

tiply

eac

h si

de b

y 4.

x!

-10

Take

the

squa

re r

oot o

f eac

h si

de.

Sinc

e lo

gari

thm

s ar

e un

defi

ned

for

x$

0,%

10 is

an

extr

aneo

us s

olut

ion.

The

onl

y so

luti

on is

10.

b.lo

g 2x

(lo

g 2(x

(2)

!3

log 2

x(

log 2

(x(

2) !

3O

rigin

al e

quat

ion

log 2

x(x

(2)

!3

Pro

duct

Pro

pert

y

x(x

(2)

!23

Def

initi

on o

f log

arith

m

x2(

2x!

8D

istr

ibut

ive

Pro

pert

y

x2(

2x %

8 !

0S

ubtr

act 8

from

eac

h si

de.

(x(

4)(x

%2)

!0

Fac

tor.

x!

2or

x!

%4

Zer

o P

rodu

ct P

rope

rty

Sinc

e lo

gari

thm

s ar

e un

defi

ned

for

x$

0,%

4 is

an

extr

aneo

us s

olut

ion.

The

onl

y so

luti

on is

2.

Sol

ve e

ach

equ

atio

n.C

hec

k y

our

solu

tion

s.

1.lo

g 54

(lo

g 52x

!lo

g 524

32.

3 lo

g 46

%lo

g 48

!lo

g 4x

27

3.lo

g 625

(lo

g 6x

!lo

g 620

44.

log 2

4 %

log 2

(x(

3) !

log 2

8"

5.lo

g 62x

%lo

g 63

!lo

g 6(x

%1)

36.

2 lo

g 4(x

(1)

!lo

g 4(1

1 %

x)2

7.lo

g 2x

%3

log 2

5 !

2 lo

g 210

12,5

008.

3 lo

g 2x

%2

log 2

5x!

210

0

9.lo

g 3(c

(3)

%lo

g 3(4

c%

1) !

log 3

510

.log

5(x

(3)

%lo

g 5(2

x%

1) !

24 % 7

8 % 19

5 % 21 & 2

x2& 4x2& 4

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Pro

pert

ies

of L

ogar

ithm

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-3

10-3

Exam

ple

Exam

ple

Exer

cises

Exer

cises


A


Skil

ls P

ract

ice

Pro

pert

ies

of L

ogar

ithm

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-3

10-3

©G

lenc

oe/M

cGra

w-H

ill58

7G

lenc

oe A

lgeb

ra 2

Lesson 10-3

Use

log

23

)1.

5850

an

d l

og2

5 )

2.32

19 t

o ap

pro

xim

ate

the

valu

e of

eac

hex

pre

ssio

n.

1.lo

g 225

4.64

382.

log 2

274.

755

3.lo

g 2"

0.73

694.

log 2

0.73

69

5.lo

g 215

3.90

696.

log 2

455.

4919

7.lo

g 275

6.22

888.

log 2

0.6

"0.

7369

9.lo

g 2"

1.58

5010

.log

20.

8481

Sol

ve e

ach

equ

atio

n.C

hec

k y

our

solu

tion

s.

11.l

og10

27 !

3 lo

g 10

x3

12.3

log 7

4 !

2 lo

g 7b

8

13.l

og4

5 (

log 4

x!

log 4

6012

14.l

og6

2c(

log 6

8 !

log 6

805

15.l

og5

y%

log 5

8 !

log 5

18

16.l

og2

q%

log 2

3 !

log 2

721

17.l

og9

4 (

2 lo

g 95

!lo

g 9w

100

18.3

log 8

2 %

log 8

4 !

log 8

b2

19.l

og10

x(

log 1

0(3

x%

5) !

log 1

02

220

.log

4x

(lo

g 4(2

x%

3) !

log 4

22

21.l

og3

d(

log 3

3 !

39

22.l

og10

y%

log 1

0(2

%y)

!0

1

23.l

og2

s(

2 lo

g 25

!0

24.l

og2

(x(

4) %

log 2

(x%

3) !

34

25.l

og4

(n(

1) %

log 4

(n%

2) !

13

26.l

og5

10 (

log 5

12 !

3 lo

g 52

(lo

g 5a

15

1 % 25

9 & 51 & 3

5 & 33 & 5

©G

lenc

oe/M

cGra

w-H

ill58

8G

lenc

oe A

lgeb

ra 2

Use

log

105

)0.

6990

an

d l

og10

7 )

0.84

51 t

o ap

pro

xim

ate

the

valu

e of

eac

hex

pre

ssio

n.

1.lo

g 10

351.

5441

2.lo

g 10

251.

3980

3.lo

g 10

0.14

614.

log 1

0"

0.14

61

5.lo

g 10

245

2.38

926.

log 1

017

52.

2431

7.lo

g 10

0.2

"0.

6990

8.lo

g 10

0.55

29

Sol

ve e

ach

equ

atio

n.C

hec

k y

our

solu

tion

s.

9.lo

g 7n

!lo

g 78

410

.log

10u

!lo

g 10

48

11.l

og6

x(

log 6

9 !

log 6

546

12.l

og8

48 %

log 8

w!

log 8

412

13.l

og9

(3u

(14

) %

log 9

5 !

log 9

2u2

14.4

log 2

x(

log 2

5 !

log 2

405

3

15.l

og3

y!

%lo

g 316

(lo

g 364

16.l

og2

d!

5 lo

g 22

%lo

g 28

4

17.l

og10

(3m

%5)

(lo

g 10

m!

log 1

02

218

.log

10(b

(3)

(lo

g 10

b!

log 1

04

1

19.l

og8

(t(

10)

%lo

g 8(t

%1)

!lo

g 812

220

.log

3(a

(3)

(lo

g 3(a

(2)

!lo

g 36

0

21.l

og10

(r(

4) %

log 1

0r

!lo

g 10

(r(

1)2

22.l

og4

(x2

%4)

%lo

g 4(x

(2)

!lo

g 41

3

23.l

og10

4 (

log 1

0w

!2

2524

.log

8(n

%3)

(lo

g 8(n

(4)

!1

4

25.3

log 5

(x2

(9)

%6

!0

*4

26.l

og16

(9x

(5)

%lo

g 16

(x2

%1)

!3

27.l

og6

(2x

%5)

(1

!lo

g 6(7

x(

10)

828

.log

2(5

y(

2) %

1 !

log 2

(1 %

2y)

0

29.l

og10

(c2

%1)

%2

!lo

g 10

(c(

1)10

130

.log

7x

(2

log 7

x%

log 7

3 !

log 7

726

31.S

OU

ND

The

loud

ness

Lof

a s

ound

in d

ecib

els

is g

iven

by

L!

10 lo

g 10

R,w

here

Ris

the

soun

d’s

rela

tive

inte

nsit

y.If

the

inte

nsit

y of

a c

erta

in s

ound

is t

ripl

ed,b

y ho

w m

any

deci

bels

doe

s th

e so

und

incr

ease

?ab

out

4.8

db

32.E

ART

HQ

UA

KES

An

eart

hqua

ke r

ated

at

3.5

on t

he R

icht

er s

cale

is f

elt

by m

any

peop

le,

and

an e

arth

quak

e ra

ted

at 4

.5 m

ay c

ause

loca

l dam

age.

The

Ric

hter

sca

le m

agni

tude

read

ing

mis

giv

en b

y m

!lo

g 10

x,w

here

xre

pres

ents

the

am

plit

ude

of t

he s

eism

ic w

ave

caus

ing

grou

nd m

otio

n.H

ow m

any

tim

es g

reat

er is

the

am

plit

ude

of a

n ea

rthq

uake

tha

tm

easu

res

4.5

on t

he R

icht

er s

cale

tha

n on

e th

at m

easu

res

3.5?

10 t

imes

1 & 2

1 % 41 & 3

3 & 22 & 3

25 & 75 & 77 & 5

Pra

ctic

e (A

vera

ge)

Pro

pert

ies

of L

ogar

ithm

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-3

10-3



Rea

din

g t

o L

earn

Math

emati

csP

rope

rtie

s of

Log

arith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-3

10-3

©G

lenc

oe/M

cGra

w-H

ill58

9G

lenc

oe A

lgeb

ra 2

Lesson 10-3

Pre-

Act

ivit

yH

ow a

re t

he

pro

per

ties

of

exp

onen

ts a

nd

log

arit

hm

s re

late

d?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 10

-3 a

t th

e to

p of

pag

e 54

1 in

you

r te

xtbo

ok.

Fin

d th

e va

lue

of lo

g 512

5.3

Fin

d th

e va

lue

of lo

g 55.

1F

ind

the

valu

e of

log 5

(125

)5)

.2

Whi

ch o

f th

e fo

llow

ing

stat

emen

ts is

tru

e?B

A.

log 5

(125

)5)

!(l

og5

125)

)(l

og5

5)B

.log

5(1

25 )

5) !

log 5

125

%lo

g 55

Rea

din

g t

he

Less

on

1.E

ach

of t

he p

rope

rtie

s of

loga

rith

ms

can

be s

tate

d in

wor

ds o

r in

sym

bols

.Com

plet

e th

est

atem

ents

of

thes

e pr

oper

ties

in w

ords

.

a.T

he lo

gari

thm

of

a qu

otie

nt is

the

of

the

loga

rith

ms

of t

he

and

the

.

b.T

he lo

gari

thm

of

a po

wer

is t

he

of t

he lo

gari

thm

of

the

base

and

the

.

c.T

he lo

gari

thm

of

a pr

oduc

t is

the

of

the

loga

rith

ms

of it

s

.

2.St

ate

whe

ther

eac

h of

the

fol

low

ing

equa

tion

s is

tru

eor

fal

se.I

f the

sta

tem

ent

is t

rue,

nam

e th

e pr

oper

ty o

f lo

gari

thm

s th

at is

illu

stra

ted.

a.lo

g 310

!lo

g 330

%lo

g 33

true

;Quo

tient

Pro

pert

yb.

log 4

12 !

log 4

4 (

log 4

8fa

lse

c.lo

g 281

!2

log 2

9tr

ue;P

ower

Pro

pert

yd.

log 8

30 !

log 8

5 '

log 8

6fa

lse

3.T

he a

lgeb

raic

pro

cess

of

solv

ing

the

equa

tion

log 2

x(

log 2

(x(

2) !

3 le

ads

to “

x!

%4

or x

!2.

”D

oes

this

mea

n th

at b

oth

%4

and

2 ar

e so

luti

ons

of t

he lo

gari

thm

ic e

quat

ion?

Exp

lain

you

r re

ason

ing.

Sam

ple

answ

er:N

o;2

is a

sol

utio

n be

caus

e it

chec

ks:l

og2

2 #

log 2

(2 #

2) !

log 2

2 #

log 2

4 !

1 #

2 !

3.H

owev

er,

beca

use

log 2

("4)

and

log 2

("2)

are

und

efin

ed,"

4 is

an

extr

aneo

usso

lutio

n an

d m

ust

be e

limin

ated

.The

onl

y so

lutio

n is

2.

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r so

met

hing

is t

o re

late

it s

omet

hing

you

alr

eady

kno

w.U

se w

ords

to e

xpla

in h

ow t

he P

rodu

ct P

rope

rty

for

expo

nent

s ca

n he

lp y

ou r

emem

ber

the

prod

uct

prop

erty

for

loga

rith

ms.

Sam

ple

answ

er:W

hen

you

mul

tiply

two

num

bers

or

expr

essi

ons

with

the

sam

e ba

se,y

ou a

ddth

e ex

pone

nts

and

keep

the

sam

e ba

se.L

ogar

ithm

s ar

e ex

pone

nts,

so t

o fin

d th

e lo

gari

thm

of

apr

oduc

t,yo

u ad

dth

e lo

gari

thm

s of

the

fac

tors

,kee

ping

the

sam

e ba

se.

fact

ors

sum

expo

nent

prod

uct

deno

min

ator

num

erat

ordi

ffer

ence

©G

lenc

oe/M

cGra

w-H

ill59

0G

lenc

oe A

lgeb

ra 2

Spi

rals

Con

side

r an

ang

le in

sta

ndar

d po

siti

on w

ith

its

vert

ex a

t a

poin

t O

calle

d th

epo

le.I

ts in

itia

l sid

e is

on

a co

ordi

nati

zed

axis

cal

led

the

pola

r ax

is.A

poi

nt P

on t

he t

erm

inal

sid

e of

the

ang

le is

nam

ed b

y th

e po

lar

coor

dina

tes

(r,.

),w

here

ris

the

dir

ecte

d di

stan

ce o

f th

e po

int

from

Oan

d .

is t

he m

easu

re o

fth

e an

gle.

Gra

phs

in t

his

syst

em m

ay b

e dr

awn

on p

olar

coo

rdin

ate

pape

rsu

ch a

s th

e ki

nd s

how

n be

low

.

1.U

se a

cal

cula

tor

to c

ompl

ete

the

tabl

e fo

r lo

g 2r

!& 12!

0&.

(Hin

t:To

fin

d !

on a

cal

cula

tor,

pres

s 12

0 r

2 .)

2.P

lot

the

poin

ts f

ound

in E

xerc

ise

1 on

the

gri

d ab

ove

and

conn

ect

to

form

a s

moo

th c

urve

.

Thi

s ty

pe o

f sp

iral

is c

alle

d a

loga

rith

mic

spi

ral b

ecau

se t

he a

ngle

m

easu

res

are

prop

orti

onal

to

the

loga

rith

ms

of t

he r

adii.

r1

23

45

67

8

!0+

120+

190+

240+

279+

310+

337+

360+

)

LOG

!)

LO

G"

01020

30

40

5060

7080

9010

011

012

013

0

140

150

160

170

180

190 200 21

0 220 23

024

025

026

027

028

029

030

031

0

32033

0340350

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-3

10-3


A


Stu

dy

Gu

ide

an

d I

nte

rven

tion

Com

mon

Log

arith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-4

10-4

©G

lenc

oe/M

cGra

w-H

ill59

1G

lenc

oe A

lgeb

ra 2

Lesson 10-4

Co

mm

on

Lo

gar

ith

ms

Bas

e 10

loga

rith

ms

are

calle

d co

mm

on l

ogar

ith

ms.

The

expr

essi

on lo

g 10

xis

usu

ally

wri

tten

wit

hout

the

sub

scri

pt a

s lo

g x.

Use

the

ke

y on

your

cal

cula

tor

to e

valu

ate

com

mon

loga

rith

ms.

The

rel

atio

n be

twee

n ex

pone

nts

and

loga

rith

ms

give

s th

e fo

llow

ing

iden

tity

.

Inve

rse

Pro

pert

y of

Log

arith

ms

and

Exp

onen

ts10

log

x!

x

Eva

luat

e lo

g 50

to

fou

r d

ecim

al p

lace

s.U

se t

he L

OG

key

on

your

cal

cula

tor.

To f

our

deci

mal

pla

ces,

log

50 !

1.69

90.

Sol

ve 3

2x#

1!

12.

32x

(1

!12

Orig

inal

equ

atio

n

log

32x

(1

!lo

g 12

Pro

pert

y of

Equ

ality

for

Loga

rithm

s

(2x

(1)

log

3 !

log

12P

ower

Pro

pert

y of

Log

arith

ms

2x(

1 !

Div

ide

each

sid

e by

log

3.

2x!

%1

Sub

trac

t 1 fr

om e

ach

side

.

x!

!%

1 "M

ultip

ly e

ach

side

by

.

x)

0.63

09

Use

a c

alcu

lato

r to

eva

luat

e ea

ch e

xpre

ssio

n t

o fo

ur

dec

imal

pla

ces.

1.lo

g 18

2.lo

g 39

3.lo

g 12

01.

2553

1.59

112.

0792

4.lo

g 5.

85.

log

42.3

6.lo

g 0.

003

0.76

341.

6263

"2.

5229

Sol

ve e

ach

equ

atio

n o

r in

equ

alit

y.R

oun

d t

o fo

ur

dec

imal

pla

ces.

7.43

x!

120.

5975

8.6x

(2

!18

"0.

3869

9.54

x%

2!

120

1.24

3710

.73x

%1

,21

{x|x

)0.

8549

}

11.2

.4x

(4

!30

"0.

1150

12.6

.52x

,20

0{x

|x)

1.41

53}

13.3

.64x

%1

!85

.41.

1180

14.2

x(

5!

3x%

213

.966

6

15.9

3x!

45x

(2

"8.

1595

16.6

x%

5!

27x

(3

"3.

6069

1 & 2lo

g 12

& log

31 & 2lo

g 12

& log

3

log

12& lo

g 3

LOG

Exer

cises

Exer

cises

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

©G

lenc

oe/M

cGra

w-H

ill59

2G

lenc

oe A

lgeb

ra 2

Ch

ang

e o

f B

ase

Form

ula

The

fol

low

ing

form

ula

is u

sed

to c

hang

e ex

pres

sion

s w

ith

diff

eren

t lo

gari

thm

ic b

ases

to

com

mon

loga

rith

m e

xpre

ssio

ns.

Cha

nge

of B

ase

Form

ula

For

all

posi

tive

num

bers

a, b

, and

n, w

here

a"

1 an

d b

"1,

log a

n!

Exp

ress

log

815

in

ter

ms

of c

omm

on l

ogar

ith

ms.

Th

en a

pp

roxi

mat

eit

s va

lue

to f

our

dec

imal

pla

ces.

log 8

15!

Cha

nge

of B

ase

For

mul

a

)1.

3023

Sim

plify

.

The

val

ue o

f lo

g 815

is a

ppro

xim

atel

y 1.

3023

.

Exp

ress

eac

h l

ogar

ith

m i

n t

erm

s of

com

mon

log

arit

hm

s.T

hen

ap

pro

xim

ate

its

valu

e to

fou

r d

ecim

al p

lace

s.

1.lo

g 316

2.lo

g 240

3.lo

g 535

,2.5

237

,5.3

219

,2.2

091

4.lo

g 422

5.lo

g 12

200

6.lo

g 250

,2.2

297

,2.1

322

,5.6

439

7.lo

g 50.

48.

log 3

29.

log 4

28.5

,"0.

5693

,0.6

309

,2.4

164

10.l

og3

(20)

211

.log

6(5

)412

.log

8(4

)5

,5.4

537

,3.5

930

,3.3

333

13.l

og5

(8)3

14.l

og2

(3.6

)615

.log

12(1

0.5)

4

,3.8

761

,11.

0880

,3.7

851

16.l

og3

#15

0$

17.l

og4

3 #39$

18.l

og5

4 #16

00$

,2.2

804

,0.8

809

,1.1

460

log

1600

%%

4 lo

g 5

log

39% 3

log

4lo

g 15

0% 2

log

3

4 lo

g 10

.5%

%lo

g 12

6 lo

g 3.

6%

%lo

g 2

3 lo

g 8

%lo

g 5

5 lo

g 4

%lo

g 8

4 lo

g 5

%lo

g 6

2 lo

g 20

%%

log

3

log

28.5

%%

log

4lo

g 2

% log

3lo

g 0.

4% lo

g 5

log

50% lo

g 2

log

200

% log

12lo

g 22

% log

4

log

35% lo

g 5

log

40% lo

g 2

log

16% lo

g 3

log 10

15& lo

g 108

log b

n& lo

g ba

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Com

mon

Log

arith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-4

10-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Com

mon

Log

arith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-4

10-4

©G

lenc

oe/M

cGra

w-H

ill59

3G

lenc

oe A

lgeb

ra 2

Lesson 10-4

Use

a c

alcu

lato

r to

eva

luat

e ea

ch e

xpre

ssio

n t

o fo

ur

dec

imal

pla

ces.

1.lo

g 6

0.77

822.

log

151.

1761

3.lo

g 1.

10.

0414

4.lo

g 0.

3"

0.52

29

Use

th

e fo

rmu

la p

H !

"lo

g[H

#]

to f

ind

th

e p

H o

f ea

ch s

ubs

tan

ce g

iven

its

con

cen

trat

ion

of

hyd

roge

n i

ons.

5.ga

stri

c ju

ices

:[H

(] !

1.0

/10

%1

mol

e pe

r lit

er1.

0

6.to

mat

o ju

ice:

[H(

] !7.

94 /

10%

5m

ole

per

liter

4.1

7.bl

ood:

[H(

] !3.

98 /

10%

8m

ole

per

liter

7.4

8.to

othp

aste

:[H

(] !

1.26

/10

%10

mol

e pe

r lit

er9.

9

Sol

ve e

ach

equ

atio

n o

r in

equ

alit

y.R

oun

d t

o fo

ur

dec

imal

pla

ces.

9.3x

#24

3{x

|x$

5}10

.16v

+&v '

v(

"(

11.8

p!

501.

8813

12.7

y!

151.

3917

13.5

3b!

106

0.96

5914

.45k

!37

0.52

09

15.1

27p

!12

00.

2752

16.9

2m!

270.

75

17.3

r%

5!

4.1

6.28

4318

.8y

(4

#15

{y|y

$"

2.69

77}

19.7

.6d

(3

!57

.2"

1.00

4820

.0.5

t%

8!

16.3

3.97

32

21.4

2x2

!84

*1.

0888

22.5

x2(

1 !10

*0.

6563

Exp

ress

eac

h l

ogar

ith

m i

n t

erm

s of

com

mon

log

arit

hm

s.T

hen

ap

pro

xim

ate

its

valu

e to

fou

r d

ecim

al p

lace

s.

23.l

og3

7;1

.771

224

.log

566

;2.6

032

25.l

og2

35;5

.129

326

.log

610

;1.2

851

log 10

10%

%lo

g 106

log 10

35%

%lo

g 102

log 10

66%

%lo

g 105

log 10

7% lo

g 103

1 % 21 & 4

©G

lenc

oe/M

cGra

w-H

ill59

4G

lenc

oe A

lgeb

ra 2

Use

a c

alcu

lato

r to

eva

luat

e ea

ch e

xpre

ssio

n t

o fo

ur

dec

imal

pla

ces.

1.lo

g 10

12.

0043

2.lo

g 2.

20.

3424

3.lo

g 0.

05"

1.30

10

Use

th

e fo

rmu

la p

H !

"lo

g[H

#]

to f

ind

th

e p

H o

f ea

ch s

ubs

tan

ce g

iven

its

con

cen

trat

ion

of

hyd

roge

n i

ons.

4.m

ilk:[

H(

] !2.

51 /

10%

7m

ole

per

liter

6.6

5.ac

id r

ain:

[H(

] !2.

51 /

10%

6m

ole

per

liter

5.6

6.bl

ack

coff

ee:[

H(

] !1.

0 /

10%

5m

ole

per

liter

5.0

7.m

ilk o

f m

agne

sia:

[H(

] !3.

16 /

10%

11m

ole

per

liter

10.5

Sol

ve e

ach

equ

atio

n o

r in

equ

alit

y.R

oun

d t

o fo

ur

dec

imal

pla

ces.

8.2x

$25

{x|x

'4.

6439

}9.

5a!

120

2.97

4610

.6z

!45

.62.

1319

11.9

m,

100

{m|m

)2.

0959

}12

.3.5

x!

47.9

3.08

8513

.8.2

y!

64.5

1.98

02

14.2

b(

1+

7.31

{b|b

(1.

8699

}15.

42x

!27

1.18

8716

.2a

%4

!82

.110

.359

3

17.9

z%

2#

38{z

|z$

3.65

55}

18.5

w(

3!

17"

1.23

9619

.30x

2!

50*

1.07

25

20.5

x2%

3!

72*

2.37

8521

.42x

!9x

(1

3.81

8822

.2n

(1

!52

n%

10.

9117

Exp

ress

eac

h l

ogar

ith

m i

n t

erm

s of

com

mon

log

arit

hm

s.T

hen

ap

pro

xim

ate

its

valu

e to

fou

r d

ecim

al p

lace

s.

23.l

og5

12;1

.544

024

.log

832

;1.6

667

25.l

og11

9 ;0

.916

3

26.l

og2

18

;4.1

699

27.l

og9

6;0

.815

528

.log

7#

8$;

29.H

ORT

ICU

LTU

RE

Sibe

rian

iris

es f

lour

ish

whe

n th

e co

ncen

trat

ion

of h

ydro

gen

ions

[H(

]in

the

soi

l is

not

less

tha

n 1.

58 /

10%

8m

ole

per

liter

.Wha

t is

the

pH

of

the

soil

in w

hich

thes

e ir

ises

will

flo

uris

h?7.

8 or

less

30.A

CID

ITY

The

pH

of

vine

gar

is 2

.9 a

nd t

he p

H o

f m

ilk is

6.6

.How

man

y ti

mes

gre

ater

isth

e hy

drog

en io

n co

ncen

trat

ion

of v

ineg

ar t

han

of m

ilk?

abou

t 50

00

31.B

IOLO

GY

The

re a

re in

itia

lly 1

000

bact

eria

in a

cul

ture

.The

num

ber

of b

acte

ria

doub

les

each

hou

r.T

he n

umbe

r of

bac

teri

a N

pres

ent

afte

r t

hour

s is

N!

1000

(2)t

.How

long

will

it t

ake

the

cult

ure

to in

crea

se t

o 50

,000

bac

teri

a?ab

out

5.6

h

32.S

OU

ND

An

equa

tion

for

loud

ness

Lin

dec

ibel

s is

giv

en b

y L

!10

log

R,w

here

Ris

the

soun

d’s

rela

tive

inte

nsit

y.A

n ai

r-ra

id s

iren

can

rea

ch 1

50 d

ecib

els

and

jet

engi

ne n

oise

can

reac

h 12

0 de

cibe

ls.H

ow m

any

tim

es g

reat

er is

the

rel

ativ

e in

tens

ity

of t

he a

ir-r

aid

sire

n th

an t

hat

of t

he je

t en

gine

noi

se?

1000

log 10

8% 2

log 10

7lo

g 106

%%

log 10

9lo

g 1018

%%

log 10

2

log 10

9%

%lo

g 1011

log 10

32%

%lo

g 108

log 10

12%

%lo

g 105

Pra

ctic

e (A

vera

ge)

Com

mon

Log

arith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-4

10-4

0.53

43


A


Rea

din

g t

o L

earn

Math

emati

csC

omm

on L

ogar

ithm

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-4

10-4

©G

lenc

oe/M

cGra

w-H

ill59

5G

lenc

oe A

lgeb

ra 2

Lesson 10-4

Pre-

Act

ivit

yW

hy

is a

log

arit

hm

ic s

cale

use

d t

o m

easu

re a

cid

ity?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 10

-4 a

t th

e to

p of

pag

e 54

7 in

you

r te

xtbo

ok.

Whi

ch s

ubst

ance

is m

ore

acid

ic,m

ilk o

r to

mat

oes?

to

mat

oes

Rea

din

g t

he

Less

on

1.R

hond

a us

ed t

he f

ollo

win

g ke

ystr

okes

to

ente

r an

exp

ress

ion

on h

er g

raph

ing

calc

ulat

or:

17

The

cal

cula

tor

retu

rned

the

res

ult

1.23

0448

921.

Whi

ch o

f th

e fo

llow

ing

conc

lusi

ons

are

corr

ect?

a,c,

and

d

a.T

he b

ase

10 lo

gari

thm

of

17 is

abo

ut 1

.230

4.

b.T

he b

ase

17 lo

gari

thm

of

10 is

abo

ut 1

.230

4.

c.T

he c

omm

on lo

gari

thm

of

17 is

abo

ut 1

.230

449.

d.10

1.23

0448

921

is v

ery

clos

e to

17.

e.T

he c

omm

on lo

gari

thm

of

17 is

exa

ctly

1.2

3044

8921

.

2.M

atch

eac

h ex

pres

sion

fro

m t

he f

irst

col

umn

wit

h an

exp

ress

ion

from

the

sec

ond

colu

mn

that

has

the

sam

e va

lue.

a.lo

g 22

ivi.

log 4

1

b.lo

g 12

iii

ii.l

og2

8

c.lo

g 31

iii

i.lo

g 10

12

d.lo

g 5v

iv.l

og5

5

e.lo

g 10

00ii

v.lo

g 0.

1

3.C

alcu

lato

rs d

o no

t ha

ve k

eys

for

find

ing

base

8 lo

gari

thm

s di

rect

ly.H

owev

er,y

ou c

an u

se

a ca

lcul

ator

to

find

log 8

20 if

you

app

ly t

he

form

ula.

Whi

ch o

f th

e fo

llow

ing

expr

essi

ons

are

equa

l to

log 8

20?

B a

nd C

A.l

og20

8B

.C

.D

.

Hel

pin

g Y

ou

Rem

emb

er

4.So

met

imes

it is

eas

ier

to r

emem

ber

a fo

rmul

a if

you

can

sta

te it

in w

ords

.Sta

te t

hech

ange

of

base

for

mul

a in

wor

ds.

Sam

ple

answ

er:T

o ch

ange

the

loga

rith

m o

f a

num

ber

from

one

bas

e to

ano

ther

,div

ide

the

log

of t

he o

rigi

nal n

umbe

rin

the

old

bas

e by

the

log

of t

he n

ew b

ase

in t

he o

ld b

ase.

log

8& lo

g 20

log

20& lo

g 8

log 10

20& lo

g 108

chan

ge o

f ba

se

1 & 5

ENTE

R)

LO

G

©G

lenc

oe/M

cGra

w-H

ill59

6G

lenc

oe A

lgeb

ra 2

The

Slid

e R

ule

Bef

ore

the

inve

ntio

n of

ele

ctro

nic

calc

ulat

ors,

com

puta

tion

s w

ere

ofte

npe

rfor

med

on

a sl

ide

rule

.A s

lide

rule

is b

ased

on

the

idea

of

loga

rith

ms.

It h

astw

o m

ovab

le r

ods

labe

led

wit

h C

and

D s

cale

s.E

ach

of t

he s

cale

s is

loga

rith

mic

.

To m

ulti

ply

2 /

3 on

a s

lide

rule

,mov

e th

e C

rod

to

the

righ

t as

sho

wn

belo

w.Y

ou c

an f

ind

2 /

3 by

add

ing

log

2 to

log

3,an

d th

e sl

ide

rule

add

s th

ele

ngth

s fo

r yo

u.T

he d

ista

nce

you

get

is 0

.778

,or

the

loga

rith

m o

f 6.

Fol

low

th

e st

eps

to m

ake

a sl

ide

rule

.

1.U

se g

raph

pap

er t

hat

has

smal

l squ

ares

,suc

h as

10

squ

ares

to

the

inch

.Usi

ng t

he s

cale

s sh

own

at

the

righ

t,pl

ot t

he c

urve

y!

log

xfo

r x

!1,

1.5,

and

the

who

le n

umbe

rs f

rom

2 t

hrou

gh 1

0.M

ake

an o

bvio

us h

eavy

dot

for

eac

h po

int

plot

ted.

2.Yo

u w

ill n

eed

two

stri

ps o

f ca

rdbo

ard.

A

5-by

-7 in

dex

card

,cut

in h

alf

the

long

way

,w

ill w

ork

fine

.Tur

n th

e gr

aph

you

mad

e in

E

xerc

ise

1 si

dew

ays

and

use

it t

o m

ark

a lo

gari

thm

ic s

cale

on

each

of

the

two

stri

ps.T

he f

igur

e sh

ows

the

mar

k fo

r 2

bein

g dr

awn.

3.E

xpla

in h

ow t

o us

e a

slid

e ru

le t

o di

vide

8 b

y 2.

Line

up

the

2 on

th

e C

sca

le w

ith t

he 8

on

the

D s

cale

.The

quo

tient

is t

he

num

ber

on t

he D

sca

le b

elow

th

e 1

on t

he C

sca

le.

0

0.1

0.2

0.3

y

1 2

11.

52

y =

log

x

0.1

0.2

12

1 21

CD

2 4

3 6

45

67

89

83

57

9

log

6

log

3lo

g 2

12

34

56

78

9

12

34

56

78

9

C D

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-4

10-4

1–2.

See

st

uden

ts’w

ork.



Stu

dy

Gu

ide

an

d I

nte

rven

tion

Bas

e e

and

Nat

ural

Log

arith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-5

10-5

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lenc

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cGra

w-H

ill59

7G

lenc

oe A

lgeb

ra 2

Lesson 10-5

Bas

e e

and

Nat

ura

l Lo

gar

ith

ms

The

irra

tion

al n

umbe

r e

)2.

7182

8… o

ften

occ

urs

as t

he b

ase

for

expo

nent

ial a

nd lo

gari

thm

ic f

unct

ions

tha

t de

scri

be r

eal-

wor

ld p

heno

men

a.

Nat

ural

Bas

e e

As

nin

crea

ses,

!1 (

"nap

proa

ches

e)

2.71

828…

.

ln x

!lo

g ex

The

fun

ctio

ns y

!ex

and

y!

ln x

are

inve

rse

func

tion

s.

Inve

rse

Pro

pert

y of

Bas

e e

and

Nat

ural

Log

arith

ms

eln

x!

xln

ex

!x

Nat

ural

bas

e ex

pres

sion

s ca

n be

eva

luat

ed u

sing

the

ex

and

ln k

eys

on y

our

calc

ulat

or.

Eva

luat

e ln

168

5.U

se a

cal

cula

tor.

ln 1

685

)7.

4295

Wri

te a

log

arit

hm

ic e

quat

ion

equ

ival

ent

to e

2x!

7.e2

x!

7 →

log e

7 !

2xor

2x

!ln

7

Eva

luat

e ln

e18

.U

se t

he I

nver

se P

rope

rty

of B

ase

ean

d N

atur

al L

ogar

ithm

s.ln

e18

!18

Use

a c

alcu

lato

r to

eva

luat

e ea

ch e

xpre

ssio

n t

o fo

ur

dec

imal

pla

ces.

1.ln

732

2.ln

84,

350

3.ln

0.7

354.

ln 1

006.

5958

11.3

427

"0.

3079

4.60

52

5.ln

0.0

824

6.ln

2.3

887.

ln 1

28,2

458.

ln 0

.006

14"

2.49

620.

8705

11.7

617

"5.

0929

Wri

te a

n e

quiv

alen

t ex

pon

enti

al o

r lo

gari

thm

ic e

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ion

.

9.e1

5!

x10

.e3x

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11.l

n 20

!x

12.l

n x

!8

ln x

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3x!

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ple1

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lenc

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cGra

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ill59

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lenc

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lgeb

ra 2

Equ

atio

ns

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ual

itie

s w

ith

ean

d ln

All

prop

erti

es o

f lo

gari

thm

s fr

omea

rlie

r le

sson

s ca

n be

use

d to

sol

ve e

quat

ions

and

ineq

ualit

ies

wit

h na

tura

l log

arit

hms.

Sol

ve e

ach

equ

atio

n o

r in

equ

alit

y.

a.3e

2x(

2 !

103e

2x(

2 !

10O

rigin

al e

quat

ion

3e2x

!8

Sub

trac

t 2 fr

om e

ach

side

.

e2x

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ivid

e ea

ch s

ide

by 3

.

ln e

2x!

ln

Pro

pert

y of

Equ

ality

for

Loga

rithm

s

2x!

ln

Inve

rse

Pro

pert

y of

Exp

onen

ts a

nd L

ogar

ithm

s

x!

ln

Mul

tiply

eac

h si

de b

y &1 2& .

x)

0.49

04U

se a

cal

cula

tor.

b.ln

(4x

%1)

$2

ln (

4x%

1) $

2O

rigin

al in

equa

lity

eln

(4x

%1)

$e2

Writ

e ea

ch s

ide

usin

g ex

pone

nts

and

base

e.

0 $

4x%

1 $

e2In

vers

e P

rope

rty

of E

xpon

ents

and

Log

arith

ms

1 $

4x$

e2(

1A

dditi

on P

rope

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s

$x

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0.25

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a c

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ve e

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1.e4

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8 & 31 & 2

8 & 38 & 3

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ide

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nte

rven

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onti

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)

Bas

e e

and

Nat

ural

Log

arith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

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RIO

D__

___

10-5

10-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises


A


NA

ME

____

____

____

____

____

____

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ATE

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10-5

10-5

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lenc

oe/M

cGra

w-H

ill59

9G

lenc

oe A

lgeb

ra 2

Lesson 10-5

Use

a c

alcu

lato

r to

eva

luat

e ea

ch e

xpre

ssio

n t

o fo

ur

dec

imal

pla

ces.

1.e3

20.0

855

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20.

1353

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20.

6931

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2.40

79

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te a

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ve e

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luat

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xpre

ssio

n t

o fo

ur

dec

imal

pla

ces.

1.e1

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4817

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INV

ESTI

NG

For

Exe

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5 an

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he

form

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d t

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AD

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The

am

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ME

____

____

____

____

____

____

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____

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____

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ATE

____

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___

10-5

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Rea

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ase

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____

____

____

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Rea

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-5 a

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p of

pag

e 55

4 in

you

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Supp

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ing

keys

trok

es in

his

cal

cula

tor:

5

The

cal

cula

tor

retu

rned

the

res

ult

1.60

9437

912.

Whi

ch o

f th

e fo

llow

ing

conc

lusi

ons

are

corr

ect?

d an

d f

a.T

he c

omm

on lo

gari

thm

of

5 is

abo

ut 1

.609

4.

b.T

he n

atur

al lo

gari

thm

of

5 is

exa

ctly

1.6

0943

7912

.

c.T

he b

ase

5 lo

gari

thm

of e

is a

bout

1.6

094.

d.T

he n

atur

al lo

gari

thm

of

5 is

abo

ut 1

.609

438.

e.10

1.60

9437

912

is v

ery

clos

e to

5.

f.e1

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ver

y cl

ose

to 5

.

2.M

atch

eac

h ex

pres

sion

fro

m t

he f

irst

col

umn

wit

h it

s va

lue

in t

he s

econ

d co

lum

n.So

me

choi

ces

may

be

used

mor

e th

an o

nce

or n

ot a

t al

l.

a.el

n 5

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som

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it t

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ne e

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ose

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stud

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wit

h a

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e w

ho is

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whe

n as

ked

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valu

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ow w

ould

you

expl

ain

to h

im a

n ea

sy w

ay t

o fi

gure

thi

s ou

t?S

ampl

e an

swer

:ln

mea

ns n

atur

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g.Th

e na

tura

l log

of

e3is

the

pow

er t

o w

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you

rai

se e

to g

et e

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isis

obv

ious

ly 3

.

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The

fol

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expr

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an b

e us

ed t

o ap

prox

imat

e e.

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reat

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are

used

,the

val

ue o

f th

e ex

pres

sion

app

roxi

mat

es e

mor

e an

dm

ore

clos

ely.

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& n1 & "n

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ther

way

to

appr

oxim

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eis

to

use

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ite

sum

.The

gre

ater

the

valu

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n,t

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*ca

n be

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e E

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se a

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wit

h an

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key

to f

ind

eto

7 d

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lace

s.2.

7182

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se t

he e

xpre

ssio

n !1

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to a

ppro

xim

ate

eto

3 d

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al p

lace

s.U

se

5,10

0,50

0,an

d 70

00 a

s va

lues

of n

.2.

488,

2.70

5,2.

716,

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8

3.U

se t

he in

finit

e su

m t

o ap

prox

imat

e e

to 3

dec

imal

pla

ces.

Use

the

who

le

num

bers

fro

m 3

thr

ough

6 a

s va

lues

of n

.2.

667,

2.70

8,2.

717,

2.71

8

4.W

hich

app

roxi

mat

ion

met

hod

appr

oach

es t

he v

alue

of e

mor

e qu

ickl

y?th

e in

finite

sum

5.U

se a

cal

cula

tor

wit

h a

*ke

y to

find

*to

7 d

ecim

al p

lace

s.3.

1415

927

6.U

se t

he in

finit

e pr

oduc

t to

app

roxi

mat

e *

to 3

dec

imal

pla

ces.

Use

the

w

hole

num

bers

fro

m 3

thr

ough

6 a

s va

lues

of n

.2.

926,

2.97

2,3.

002,

3.02

3

7.D

oes

the

infin

ite

prod

uct

give

goo

d ap

prox

imat

ions

for

*qu

ickl

y?no

8.Sh

ow t

hat

*4

(*

5is

equ

al t

o e6

to 4

dec

imal

pla

ces.

To 4

dec

imal

pla

ces,

they

bot

h eq

ual 4

03.4

288.

9.W

hich

is la

rger

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or *

e ?e&

> &

e

10.T

he e

xpre

ssio

n x

reac

hes

a m

axim

um v

alue

at

x!

e.U

se t

his

fact

to

prov

e th

e in

equa

lity

you

foun

d in

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e 9.

e%1 e%>

&% &1 % ;#

e%1 e% $&e

> # &

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rich

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t

NA

ME

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____

____

____

____

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____

____

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RIO

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10-5

10-5


A


Stu

dy

Gu

ide

an

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Exp

onen

tial G

row

th a

nd D

ecay

NA

ME

____

____

____

____

____

____

____

____

____

____

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ATE

____

____

____

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RIO

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10-6

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Lesson 10-6

Exp

on

enti

al D

ecay

Dep

reci

atio

n of

val

ue a

nd r

adio

acti

ve d

ecay

are

exa

mpl

es o

fex

pon

enti

al d

ecay

.Whe

n a

quan

tity

dec

reas

es b

y a

fixe

d pe

rcen

t ea

ch t

ime

peri

od,t

heam

ount

of

the

quan

tity

aft

er t

tim

e pe

riod

s is

giv

en b

y y

!a(

1 %

r)t ,

whe

re a

is t

he in

itia

lam

ount

and

ris

the

per

cent

dec

reas

e ex

pres

sed

as a

dec

imal

.A

noth

er e

xpon

enti

al d

ecay

mod

el o

ften

use

d by

sci

enti

sts

is y

!ae

%kt

,whe

re k

is a

con

stan

t.

CO

NSU

MER

PR

ICES

As

tech

nol

ogy

adva

nce

s,th

e p

rice

of

man

yte

chn

olog

ical

dev

ices

su

ch a

s sc

ien

tifi

c ca

lcu

lato

rs a

nd

cam

cord

ers

goes

dow

n.

On

e br

and

of

han

d-h

eld

org

aniz

er s

ells

for

$89

.

a.If

its

pri

ce d

ecre

ases

by

6% p

er y

ear,

how

mu

ch w

ill

it c

ost

afte

r 5

year

s?U

se t

he e

xpon

enti

al d

ecay

mod

el w

ith

init

ial a

mou

nt $

89,p

erce

nt d

ecre

ase

0.06

,and

tim

e 5

year

s.y

!a(

1 %

r)t

Exp

onen

tial d

ecay

form

ula

y!

89(1

%0.

06)5

a!

89, r

!0.

06, t

!5

y!

$65.

32A

fter

5 y

ears

the

pri

ce w

ill b

e $6

5.32

.

b.A

fter

how

man

y ye

ars

wil

l it

s p

rice

be

$50?

To fi

nd w

hen

the

pric

e w

ill b

e $5

0,ag

ain

use

the

expo

nent

ial d

ecay

form

ula

and

solv

e fo

r t.

y!

a(1

%r)

tE

xpon

entia

l dec

ay fo

rmul

a

50 !

89(1

%0.

06)t

y!

50, a

!89

, r!

0.06

!(0

.94)

tD

ivid

e ea

ch s

ide

by 8

9.

log

!"!

log

(0.9

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Pro

pert

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ality

for

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rithm

s

log

!"!

tlo

g 0.

94P

ower

Pro

pert

y

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Div

ide

each

sid

e by

log

0.94

.

t)

9.3

The

pri

ce w

ill b

e $5

0 af

ter

abou

t 9.

3 ye

ars.

1.B

USI

NES

SA

fur

nitu

re s

tore

is c

losi

ng o

ut it

s bu

sine

ss.E

ach

wee

k th

e ow

ner

low

ers

pric

es b

y 25

%.A

fter

how

man

y w

eeks

will

the

sal

e pr

ice

of a

$50

0 it

em d

rop

belo

w $

100?

6 w

eeks

CA

RB

ON

DA

TIN

GU

se t

he

form

ula

y!

ae"

0.00

012t

,wh

ere

ais

th

e in

itia

l am

oun

t of

Car

bon

-14,

tis

th

e n

um

ber

of y

ears

ago

th

e an

imal

liv

ed,a

nd

yis

th

e re

mai

nin

gam

oun

t af

ter

tye

ars.

2.H

ow o

ld is

a fo

ssil

rem

ain

that

has

lost

95%

of i

ts C

arbo

n-14

?ab

out 2

5,00

0 ye

ars

old

3.H

ow o

ld is

a s

kele

ton

that

has

95%

of i

ts C

arbo

n-14

rem

aini

ng?

abou

t 427

.5 y

ears

old

log

!&5 80 9&"

&&

log

0.94

50 & 8950 & 8950 & 89

Exam

ple

Exam

ple

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cises

Exer

cises

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Exp

on

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al G

row

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pula

tion

incr

ease

and

gro

wth

of

bact

eria

col

onie

s ar

e ex

ampl

esof

exp

onen

tial

gro

wth

.Whe

n a

quan

tity

incr

ease

s by

a fi

xed

perc

ent

each

tim

e pe

riod

,the

amou

nt o

f th

at q

uant

ity

afte

r t

tim

e pe

riod

s is

giv

en b

y y

!a(

1 (

r)t ,

whe

re a

is t

he in

itia

lam

ount

and

ris

the

per

cent

incr

ease

(or

rat

e of

gro

wth

) ex

pres

sed

as a

dec

imal

.A

noth

er e

xpon

enti

al g

row

th m

odel

oft

en u

sed

by s

cien

tist

s is

y!

aekt

,whe

re k

is a

con

stan

t.

A c

omp

ute

r en

gin

eer

is h

ired

for

a s

alar

y of

$28

,000

.If

she

gets

a5%

rai

se e

ach

yea

r,af

ter

how

man

y ye

ars

wil

l sh

e be

mak

ing

$50,

000

or m

ore?

Use

the

exp

onen

tial

gro

wth

mod

el w

ith

a!

28,0

00,y

!50

,000

,and

r!

0.05

and

sol

ve f

or t

.

y!

a(1

(r)

tE

xpon

entia

l gro

wth

form

ula

50,0

00 !

28,0

00(1

(0.

05)t

y!

50,0

00, a

!28

,000

, r!

0.05

!(1

.05)

tD

ivid

e ea

ch s

ide

by 2

8,00

0.

log

!"!

log

(1.0

5)t

Pro

pert

y of

Equ

ality

of L

ogar

ithm

s

log

!"!

tlo

g 1.

05P

ower

Pro

pert

y

t!

Div

ide

each

sid

e by

log

1.05

.

t)

11.9

yea

rsU

se a

cal

cula

tor.

If r

aise

s ar

e gi

ven

annu

ally

,she

will

be

mak

ing

over

$50

,000

in 1

2 ye

ars.

1.B

AC

TER

IA G

RO

WTH

A c

erta

in s

trai

n of

bac

teri

a gr

ows

from

40

to 3

26 in

120

min

utes

.F

ind

kfo

r th

e gr

owth

for

mul

a y

!ae

kt,w

here

tis

in m

inut

es.

abou

t 0.

0175

2.IN

VES

TMEN

TC

arl p

lans

to

inve

st $

500

at 8

.25%

inte

rest

,com

poun

ded

cont

inuo

usly

.H

ow lo

ng w

ill it

tak

e fo

r hi

s m

oney

to

trip

le?

abou

t 14

yea

rs

3.SC

HO

OL

POPU

LATI

ON

The

re a

re c

urre

ntly

850

stu

dent

s at

the

hig

h sc

hool

,whi

chre

pres

ents

ful

l cap

acit

y.T

he t

own

plan

s an

add

itio

n to

hou

se 4

00 m

ore

stud

ents

.If

the

scho

ol p

opul

atio

n gr

ows

at 7

.8%

per

yea

r,in

how

man

y ye

ars

will

the

new

add

itio

nbe

ful

l?ab

out

5 ye

ars

4.EX

ERC

ISE

Hug

o be

gins

a w

alki

ng p

rogr

am b

y w

alki

ng

mile

per

day

for

one

wee

k.

Eac

h w

eek

ther

eaft

er h

e in

crea

ses

his

mile

age

by 1

0%.A

fter

how

man

y w

eeks

is h

ew

alki

ng m

ore

than

5 m

iles

per

day?

24 w

eeks

5.V

OC

AB

ULA

RY G

RO

WTH

Whe

n E

mily

was

18

mon

ths

old,

she

had

a 10

-wor

dvo

cabu

lary

.By

the

tim

e sh

e w

as 5

yea

rs o

ld (6

0 m

onth

s),h

er v

ocab

ular

y w

as 2

500

wor

ds.

If h

er v

ocab

ular

y in

crea

sed

at a

con

stan

t pe

rcen

t pe

r m

onth

,wha

t w

as t

hat

incr

ease

?ab

out 1

4%

1 & 2

log

!&5 20 8&"

& log

1.05

50 & 2850 & 2850 & 28

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Exp

onen

tial G

row

th a

nd D

ecay

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-6

10-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Exp

onen

tial G

row

th a

nd D

ecay

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-6

10-6

©G

lenc

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ill60

5G

lenc

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lgeb

ra 2

Lesson 10-6

Sol

ve e

ach

pro

blem

.

1.FI

SHIN

GIn

an

over

-fis

hed

area

,the

cat

ch o

f a

cert

ain

fish

is d

ecre

asin

g at

an

aver

age

rate

of

8% p

er y

ear.

If t

his

decl

ine

pers

ists

,how

long

will

it t

ake

for

the

catc

h to

rea

chha

lf o

f th

e am

ount

bef

ore

the

decl

ine?

abou

t 8.

3 yr

2.IN

VES

TIN

GA

lex

inve

sts

$200

0 in

an

acco

unt

that

has

a 6

% a

nnua

l rat

e of

gro

wth

.To

the

near

est

year

,whe

n w

ill t

he in

vest

men

t be

wor

th $

3600

?10

yr

3.PO

PULA

TIO

NA

cur

rent

cen

sus

show

s th

at t

he p

opul

atio

n of

a c

ity

is 3

.5 m

illio

n.U

sing

the

form

ula

P!

aert

,fin

d th

e ex

pect

ed p

opul

atio

n of

the

cit

y in

30

year

s if

the

gro

wth

rate

rof

the

pop

ulat

ion

is 1

.5%

per

yea

r,a

repr

esen

ts t

he c

urre

nt p

opul

atio

n in

mill

ions

,an

d t

repr

esen

ts t

he t

ime

in y

ears

.ab

out

5.5

mill

ion

4.PO

PULA

TIO

NT

he p

opul

atio

n P

in t

hous

ands

of

a ci

ty c

an b

e m

odel

ed b

y th

e eq

uati

onP

!80

e0.0

15t ,

whe

re t

is t

he t

ime

in y

ears

.In

how

man

y ye

ars

will

the

pop

ulat

ion

of t

heci

ty b

e 12

0,00

0?ab

out

27 y

r

5.B

AC

TER

IAH

ow m

any

days

will

it t

ake

a cu

ltur

e of

bac

teri

a to

incr

ease

fro

m 2

000

to50

,000

if t

he g

row

th r

ate

per

day

is 9

3.2%

?ab

out

4.9

days

6.N

UC

LEA

R P

OW

ERT

he e

lem

ent

plut

oniu

m-2

39 is

hig

hly

radi

oact

ive.

Nuc

lear

rea

ctor

sca

n pr

oduc

e an

d al

so u

se t

his

elem

ent.

The

hea

t th

at p

luto

nium

-239

em

its

has

help

ed t

opo

wer

equ

ipm

ent

on t

he m

oon.

If t

he h

alf-

life

of p

luto

nium

-239

is 2

4,36

0 ye

ars,

wha

t is

the

valu

e of

kfo

r th

is e

lem

ent?

abou

t 0.

0000

2845

7.D

EPR

ECIA

TIO

NA

Glo

bal P

osit

ioni

ng S

atel

lite

(GP

S) s

yste

m u

ses

sate

llite

info

rmat

ion

to lo

cate

gro

und

posi

tion

.Abu

’s s

urve

ying

fir

m b

ough

t a

GP

S sy

stem

for

$12

,500

.The

GP

S de

prec

iate

d by

a f

ixed

rat

e of

6%

and

is n

ow w

orth

$86

00.H

ow lo

ng a

go d

id A

bubu

y th

e G

PS

syst

em?

abou

t 6.

0 yr

8.B

IOLO

GY

In a

labo

rato

ry,a

n or

gani

sm g

row

s fr

om 1

00 t

o 25

0 in

8 h

ours

.Wha

t is

the

hour

ly g

row

th r

ate

in t

he g

row

th f

orm

ula

y!

a(1

(r)

t ?ab

out

12.1

3%

©G

lenc

oe/M

cGra

w-H

ill60

6G

lenc

oe A

lgeb

ra 2

Sol

ve e

ach

pro

blem

.

1.IN

VES

TIN

GT

he fo

rmul

a A

!P!1

("2t

give

s th

e va

lue

of a

n in

vest

men

t af

ter

tye

ars

in

an a

ccou

nt t

hat

earn

s an

ann

ual i

nter

est

rate

rco

mpo

unde

d tw

ice

a ye

ar.S

uppo

se $

500

is in

vest

ed a

t 6%

ann

ual i

nter

est

com

poun

ded

twic

e a

year

.In

how

man

y ye

ars

will

the

inve

stm

ent

be w

orth

$10

00?

abou

t 11

.7 y

r

2.B

AC

TER

IAH

ow m

any

hour

s w

ill it

tak

e a

cult

ure

of b

acte

ria

to in

crea

se f

rom

20

to20

00 if

the

gro

wth

rat

e pe

r ho

ur is

85%

?ab

out

7.5

h

3.R

AD

IOA

CTI

VE

DEC

AY

A r

adio

acti

ve s

ubst

ance

has

a h

alf-

life

of 3

2 ye

ars.

Fin

d th

eco

nsta

nt k

in t

he d

ecay

for

mul

a fo

r th

e su

bsta

nce.

abou

t 0.

0216

6

4.D

EPR

ECIA

TIO

NA

pie

ce o

f m

achi

nery

val

ued

at $

250,

000

depr

ecia

tes

at a

fix

ed r

ate

of12

% p

er y

ear.

Aft

er h

ow m

any

year

s w

ill t

he v

alue

hav

e de

prec

iate

d to

$10

0,00

0?ab

out

7.2

yr

5.IN

FLA

TIO

NFo

r D

ave

to b

uy a

new

car

com

para

bly

equi

pped

to

the

one

he b

ough

t 8

year

sag

o w

ould

cos

t $1

2,50

0.Si

nce

Dav

e bo

ught

the

car

,the

infl

atio

n ra

te f

or c

ars

like

his

has

been

at

an a

vera

ge a

nnua

l rat

e of

5.1

%.I

f D

ave

orig

inal

ly p

aid

$840

0 fo

r th

e ca

r,ho

wlo

ng a

go d

id h

e bu

y it

?ab

out

8 yr

6.R

AD

IOA

CTI

VE

DEC

AY

Cob

alt,

an e

lem

ent

used

to

mak

e al

loys

,has

sev

eral

isot

opes

.O

ne o

f th

ese,

coba

lt-6

0,is

rad

ioac

tive

and

has

a h

alf-

life

of 5

.7 y

ears

.Cob

alt-

60 is

use

d to

trac

e th

e pa

th o

f no

nrad

ioac

tive

sub

stan

ces

in a

sys

tem

.Wha

t is

the

val

ue o

f kfo

rC

obal

t-60

?ab

out

0.12

16

7.W

HA

LES

Mod

ern

wha

les

appe

ared

5%

10 m

illio

n ye

ars

ago.

The

ver

tebr

ae o

f a

wha

ledi

scov

ered

by

pale

onto

logi

sts

cont

ain

roug

hly

0.25

% a

s m

uch

carb

on-1

4 as

the

y w

ould

have

con

tain

ed w

hen

the

wha

le w

as a

live.

How

long

ago

did

the

wha

le d

ie?

Use

k

!0.

0001

2.ab

out

50,0

00 y

r

8.PO

PULA

TIO

NT

he p

opul

atio

n of

rab

bits

in a

n ar

ea is

mod

eled

by

the

grow

th e

quat

ion

P(t

) !8e

0.26

t ,w

here

Pis

in t

hous

ands

and

tis

in y

ears

.How

long

will

it t

ake

for

the

popu

lati

on t

o re

ach

25,0

00?

abou

t 4.

4 yr

9.D

EPR

ECIA

TIO

NA

com

pute

r sy

stem

dep

reci

ates

at

an a

vera

ge r

ate

of 4

% p

er m

onth

.If

the

valu

e of

the

com

pute

r sy

stem

was

ori

gina

lly $

12,0

00,i

n ho

w m

any

mon

ths

is it

wor

th $

7350

?ab

out

12 m

o

10.B

IOLO

GY

In a

labo

rato

ry,a

cul

ture

incr

ease

s fr

om 3

0 to

195

org

anis

ms

in 5

hou

rs.

Wha

t is

the

hou

rly

grow

th r

ate

in t

he g

row

th f

orm

ula

y!

a(1

(r)

t ?ab

out

45.4

%

r & 2

Pra

ctic

e (A

vera

ge)

Exp

onen

tial G

row

th a

nd D

ecay

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-6

10-6


A


Rea

din

g t

o L

earn

Math

emati

csE

xpon

entia

l Gro

wth

and

Dec

ay

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-6

10-6

©G

lenc

oe/M

cGra

w-H

ill60

7G

lenc

oe A

lgeb

ra 2

Lesson 10-6

Pre-

Act

ivit

yH

ow c

an y

ou d

eter

min

e th

e cu

rren

t va

lue

of y

our

car?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 10

-6 a

t th

e to

p of

pag

e 56

0 in

you

r te

xtbo

ok.

•B

etw

een

whi

ch t

wo

year

s sh

own

in t

he t

able

did

the

car

dep

reci

ate

byth

e gr

eate

st a

mou

nt?

betw

een

year

s 0

and

1•

Des

crib

e tw

o w

ays

to c

alcu

late

the

val

ue o

f th

e ca

r 6

year

s af

ter

it w

aspu

rcha

sed.

(Do

not

actu

ally

cal

cula

te t

he v

alue

.)S

ampl

e an

swer

:1.M

ultip

ly $

9200

.66

by 0

.16

and

subt

ract

the

resu

lt fr

om $

9200

.66.

2.M

ultip

ly $

9200

.66

by 0

.84.

Rea

din

g t

he

Less

on

1.St

ate

whe

ther

eac

h si

tuat

ion

is a

n ex

ampl

e of

exp

onen

tial

gro

wth

or d

ecay

.

a.A

cit

y ha

d 42

,000

res

iden

ts in

198

0 an

d 12

8,00

0 re

side

nts

in 2

000.

grow

th

b.R

aul c

ompa

red

the

valu

e of

his

car

whe

n he

bou

ght

it n

ew t

o th

e va

lue

whe

n he

trad

ed ‘;

lpit

in s

ix y

ears

late

r.de

cay

c.A

pal

eont

olog

ist

com

pare

d th

e am

ount

of

carb

on-1

4 in

the

ske

leto

n of

an

anim

alw

hen

it d

ied

to t

he a

mou

nt 3

00 y

ears

late

r.de

cay

d.M

aria

dep

osit

ed $

750

in a

sav

ings

acc

ount

pay

ing

4.5%

ann

ual i

nter

est

com

poun

ded

quar

terl

y.Sh

e di

d no

t m

ake

any

wit

hdra

wal

s or

fur

ther

dep

osit

s.Sh

e co

mpa

red

the

bala

nce

in h

er p

assb

ook

imm

edia

tely

aft

er s

he o

pene

d th

e ac

coun

t to

the

bal

ance

3

year

s la

ter.

grow

th

2.St

ate

whe

ther

eac

h eq

uati

on r

epre

sent

s ex

pone

ntia

l gro

wth

or

deca

y.

a.y

!5e

0.15

tgr

owth

b.y

!10

00(1

%0.

05)t

deca

y

c.y

!0.

3e%

1200

tde

cay

d.y

!2(

1 (

0.00

01)t

grow

th

Hel

pin

g Y

ou

Rem

emb

er

3.V

isua

lizin

g th

eir

grap

hs is

oft

en a

goo

d w

ay t

o re

mem

ber

the

diff

eren

ce b

etw

een

mat

hem

atic

al e

quat

ions

.How

can

you

r kn

owle

dge

of t

he g

raph

s of

exp

onen

tial

equ

atio

nsfr

om L

esso

n 10

-1 h

elp

you

to r

emem

ber

that

equ

atio

ns o

f th

e fo

rm y

!a(

1 (

r)t

repr

esen

t ex

pone

ntia

l gro

wth

,whi

le e

quat

ions

of

the

form

y!

a(1

%r)

tre

pres

ent

expo

nent

ial d

ecay

?S

ampl

e an

swer

:If

a$

0,th

e gr

aph

of y

!ab

xis

alw

ays

incr

easi

ng if

b

$1

and

is a

lway

s de

crea

sing

if 0

'b

'1.

Sin

ce r

is a

lway

s a

posi

tive

num

ber,

if b

!1

#r,

the

base

will

be

grea

ter

than

1 a

nd t

he f

unct

ion

will

be in

crea

sing

(gr

owth

),w

hile

if b

!1

"r,

the

base

will

be

less

tha

n 1

and

the

func

tion

will

be

decr

easi

ng (

deca

y).

©G

lenc

oe/M

cGra

w-H

ill60

8G

lenc

oe A

lgeb

ra 2

Eff

ectiv

e A

nnua

l Yie

ldW

hen

inte

rest

is c

ompo

unde

d m

ore

than

onc

e pe

r ye

ar,t

he e

ffec

tive

ann

ual

yiel

d is

hig

her

than

the

ann

ual i

nter

est

rate

.The

eff

ecti

ve a

nnua

l yie

ld,E

,is

the

inte

rest

rat

e th

at w

ould

giv

e th

e sa

me

amou

nt o

f in

tere

st if

the

inte

rest

wer

e co

mpo

unde

d on

ce p

er y

ear.

If P

dolla

rs a

re in

vest

ed f

or o

ne y

ear,

the

valu

e of

the

inve

stm

ent

at t

he e

nd o

f th

e ye

ar is

A!

P(1

(E

).If

Pdo

llars

are

inve

sted

for

one

yea

r at

a n

omin

al r

ate

rco

mpo

unde

d n

tim

es p

er y

ear,

the

valu

e of

the

inve

stm

ent

at t

he e

nd o

f th

e ye

ar is

A!

P!1

(& nr & "n .S

etti

ng

the

amou

nts

equa

l and

sol

ving

for

Ew

ill p

rodu

ce a

for

mul

a fo

r th

e ef

fect

ive

annu

al y

ield

.

P(1

(E

) !P!1

(& nr & "n

1 (

E!

!1 (

& nr & "n

E!

!1 (

& nr & "n%

1

If c

ompo

undi

ng is

con

tinu

ous,

the

valu

e of

the

inve

stm

ent

at t

he e

nd o

f on

eye

ar is

A!

Per .

Aga

in s

et t

he a

mou

nts

equa

l and

sol

ve fo

r E

.A fo

rmul

a fo

rth

e ef

fect

ive

annu

al y

ield

und

er c

onti

nuou

s co

mpo

undi

ng is

obt

aine

d.

P(1

(E

) !Pe

r

1 (

E!

er

E!

er%

1

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

10-6

10-6

Fin

d t

he

effe

ctiv

ean

nu

al y

ield

of

an i

nve

stm

ent

mad

e at

7.5%

com

pou

nd

ed m

onth

ly.

r!

0.07

5n

!12

E!

!1 (

&0.10 275 &

"12%

1 *

7.7

6%

Fin

d t

he

effe

ctiv

ean

nu

al y

ield

of

an i

nve

stm

ent

mad

e at

6.25

% c

omp

oun

ded

con

tin

uou

sly.

r!

0.06

25E

!e0

.062

5%

1 *

6.4

5%

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Fin

d t

he

effe

ctiv

e an

nu

al y

ield

for

eac

h i

nve

stm

ent.

1.10

% c

ompo

unde

d qu

arte

rly

10.3

8%2.

8.5%

com

poun

ded

mon

thly

8.84

%

3.9.

25%

com

poun

ded

cont

inuo

usly

9.69

%4.

7.75

% c

ompo

unde

d co

ntin

uous

ly8.

06%

5.6.

5% c

ompo

unde

d da

ily (

assu

me

a 36

5-da

y ye

ar)

6.72

%

6.W

hich

inve

stm

ent

yiel

ds m

ore

inte

rest

—9%

com

poun

ded

cont

inuo

usly

or

9.2%

com

poun

ded

quar

terl

y?9.

2% q

uart

erly

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