MATH 30-2

EXPONENTS & LOGARITHMS Module Seven

Module 7 - Assignment Booklet

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Lesson 1: Characteristics of Exponential Functions1. Graph each exponential function. State the domain, range, and x- and y-intercepts for

each; and tell whether each graph is increasing or decreasing.

a.

b. y = 4x

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2. Without graphing, predict the x- and y-intercepts, domain, and range for the following exponential equations. Also tell whether each graph is increasing or decreasing.

a. y = 5(1.2)x

b.

3. Complete “Your Turn” on page 345 of the textbook.

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Lesson 2: Solving Exponential Equations1. Write the expressions 36x and 2162x + 3 so that they have the same base.

2. Solve the exponential equation 32x 83x 4 by changing the bases and then using algebra. Check your answer by graphing. Be sure to include a sketch of your graph.

3. Create your own exponential equation that contains different bases. Include a full solution to your equation.

4. Trinh has decided to save money to buy a new computer. She invests $800 in an account that earns 5% per year, compounded quarterly. Determine how long she needs to invest

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her money to have $1000 by using the equation 1000 800(1.0125)n, where n is measured in quarters of a year (three-month increments).

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Lesson 3: Converting Between Logarithms and Exponents1. Convert each of the following to exponential notation.

a. y log 6000

b.

2. Convert each of the following to logarithmic notation.

a. 56 15 625

b. nt h

3. Evaluate each of the following logarithms.

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a.

b.

c.

4. a. Determine the pH of a solution with a hydrogen ion concentration of 6.37 10–9.

c. Determine the hydrogen ion concentration of a solution with a pH of 4.1.

5. Describe how the ability to change an equation from logarithmic form to exponential form can help you determine the value of a logarithm. Include an example with your explanation.

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Lesson 4: Characteristics of Logarithmic Functions1. a. Describe the characteristics of the graph of y 4 log x.

b. Graph y 4 log x using technology to check your predictions. Include a screen shot or sketch of your graph.

2. Describe at least three similarities and three differences between the functions y 10x and y log x.

3. A researcher has determined that an equation that models the height of a particular species of tree is y = 20.2 log(x), where y is height in metres and x is time in years.

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a. Graph the equation using technology. Include the sketch below.

b. Predict the height of the tree after 30 years.

c. Predict how old the tree will be when it is 15 m tall.

d. Describe how the height of the tree changes as time increases. Be specific.

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Lesson 5: Laws of Logarithms 1. Fully simplify each of the following using the laws of logarithms. Be sure to show all steps.

a.

b.

c. log x log y log z

2. Write each of the following as a sum or a difference of logarithms.

a.

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b.

3. Each of the following has an error. Identify the error and explain why it is wrong.

a.

b.

4. In the Focus section you were introduced to the equation Use the

laws of logarithms to simplify the expression

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Lesson 6: Solving Exponential Equations Using Logarithms 1. Solve 124 3(4)2x – 1 for x using logarithms.

2. Solve 12t –4 17–3t for t using logarithms.

3. Rewrite using base-10 logarithms. Find an approximate answer, rounded to three decimal places.

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4. Gallium-67 is used in nuclear medicine to help doctors locate inflammation and chronic infections. The patient is injected with a tracer (trace amount) that includes gallium-67, which collects in areas of inflammation and infection. The gallium-67 emits radiation that a special camera can detect. Gallium-67 has a half-life of 3.26 days

a. Give an exponential equation to represent the percentage of the original gallium-67 after t days.

b. Determine the amount of gallium-67 left after 4 days.

c. Solve your equation to determine the time it will take for there to be 1% of the original gallium-67.

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Lesson 7: Modelling Data Using Exponential and Logarithmic Regressions1. In 2008, Marianna paid $14 500 for a used 2005 van. She has been tracking the selling

price of the van as time has passed to decide how long to keep the van before replacing it. The data she has collected is shown in the following table:

Age (in years) 3 4 5 6 7 8

Selling Price ($) 14 500 11 600 8700 6800 4300 3500

a. Determine the equation of the exponential regression function that models the depreciation of the van.

b. What was the initial selling price of the van in 2005?

c. What is the annual depreciation rate of the van?

2. In a study, researchers found that the arterial blood pressure, P, in millimetres of mercury (mmHg) is related to the volume, V, of the blood vessel in microlitres (L). The data gathered is listed in the following table:

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Arterial Blood Pressure, P (mmHg) Blood Vessel Volume, V (L)

70 0.63

100 0.80

120 0.86

140 0.90

160 0.94

a. Using a logarithmic regression, determine the equation that represents the relationship between blood pressure, P, and the blood-vessel volume, V.

b. Predict the blood-vessel volume when the arterial blood pressure is 130 mmHg. Make sure to round your answer to the nearest hundredth of a microlitre.

c. Predict the arterial blood pressure when the vessel volume is 0.7 L. Round your answer to the nearest millimetre of mercury.

Module 7 SUMMARYIn this module you investigated the following question: How can exponents and logarithms be used to solve problems that involve growth and decay?

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Exponents and logarithms can be used to model growth, such as the rate a tree grows, and decay, such as the half-life of a radioactive element. Exponents and logarithms can both be useful when working with very large or very small values, such as pH or earthquake magnitude.

You learned about the characteristics of an exponential function and how exponential functions can be used to solve problems. You also learned about the characteristics of logarithmic functions, how logarithms can be used to solve problems, and how logarithmic scales can be used for data with an extreme range of values. You used the laws of logarithms to manipulate expressions with logarithms and to solve exponential equations. Finally, you determined regression equations for data that was logarithmic and exponential.

left: Comstock/Thinkstock; right: © Schlierner /32104381/Fotolia

In the Module 7 Project: At the Movies, you described exponential and logarithmic graphs, solved various exponential and logarithmic problems in context, and applied your knowledge of regressions to information about your favourite movie.

The following chart lists some of the key ideas that you learned in each lesson.

Lesson 1

Exponential functions can be used to model growth and decay.

Lesson 2 Exponential equations can be solved using common bases or graphically.

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Lesson 3

The expression y = ax ↔ x = loga y can be used to convert between exponential and logarithmic forms of an equation.

Source: Pre-Calculus 12. Whitby, ON: McGraw-Hill Ryerson, 2011. Reproduced with permission.

Lesson 4

Logarithmic functions can be increasing or decreasing.

Lesson 5 The laws of logarithms (shown below) can be used to change logarithmic

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expressions.

logb M + logb N = logb(M × N)

logb (MN) = n logb M

Lesson 6

Exponential equations can be solved by taking the logarithm of both sides of the equation and then using the power law of logarithms to remove the exponent.

Lesson 7

Regression equations can be determined for exponential and logarithmic data. The natural logarithm, ln, is often used in logarithmic regression equations, as in ln x = loge x, where e ≈ 2.72.

Mathematics 30-2 Learn EveryWare © 2012 Alberta EducationLast modified: Monday, 14

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MODULE 7 –EXPONENTS & LOGARITHMS SUMMATIVE ASSIGNMENTComplete the following questions from your text book. Show steps completely and clearly, as marks are assigned for mathematical literacy and communication. Always use graph paper, rulers, and pencils as necessary. Attach questions and study notes securely to this booklet before you hand everything in.

Text: Principles of Mathematics 12

Chapter 6 – EXPONENTIAL FUNCTIONS

Chapter 7 – LOGARITHMIC FUNCTIONS

Section 6.1: Page 337: #1

Section 6.2: Pages 347 to 351: #2d, 5, 8ad, 10ab

Section 6.3: Pages 361 to 365: #4ad, 7df, 9, 11

Section 6.4: Pages 377 to 385: #5, 12, 15

Section 6.5: Pages 395 to 399: #3, 4, 7, 10ab, 11ab

Section 7.1: Pages 421 to 425: #5, 7, 12

Section 7.2: Pages 436 to 438: #6ac, 8ac, 10ac, 14ac

Section 7.3: Pages 446 to 448: #6ab, 8a, 11ab, 14

Section 7.4: Pages 455 to 459: #1ab, 6ac, 8, 14, 15ab

Section 7.5: Pages 466 to 472: #3, 6

Module 7 is now complete.

Once you have received your corrected work, review your instructor’s comments and prepare for your module seven test.