Chapter 3 Exponential and Logarithmic Functions 329.pdf · graph exponential functions. I. Exponential Functions (Page 180) Polynomial functions and rational functions are examples

Name ___________________________________________________________ Date ____________

Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAE Copyright © Cengage Learning. All rights reserved. 41

Chapter 3 Exponential and Logarithmic Functions Section 3.1 Exponential Functions and Their Graphs Objective: In this lesson you learned how to recognize, evaluate, and

graph exponential functions. I. Exponential Functions (Page 180) Polynomial functions and rational functions are examples of

algebraic functions.

The exponential function f with base a is denoted by

f(x) = ax , where a > 0, a 1, and x is any real

number.

Example 1: Use a calculator to evaluate the expression 5/35 . 2.626527804 II. Graphs of Exponential Functions (Pages 181 183) For a > 1, is the graph of ( ) xf x a increasing or decreasing

over its domain? Increasing

For a > 1, is the graph of ( ) xg x a increasing or decreasing

over its domain? Decreasing

For the graph of xay or xay , a > 1, the domain is

( , ) , the range is (0, ) , and

the intercept is (0, 1) . Also, both graphs have

the x-axis as a horizontal asymptote.

Example 2: Sketch the graph of the function xxf 3)( .

Important Vocabulary Define each term or concept. Transcendental functions Functions that are not algebraic. Natural base e The irrational number e 2.718281828 . . .

What you should learn How to recognize and evaluate exponential functions with base a

What you should learn How to graph exponential functions with base a

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42 Chapter 3 Exponential and Logarithmic Functions

Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAE Copyright © Cengage Learning. All rights reserved.

III. The Natural Base e (Pages 184 185) The natural exponential function is given by the function f(x) = ex . Example 3: Use a calculator to evaluate the expression 5/3e . 1.8221188 For the graph of xexf )( , the domain is ( , ) ,

the range is (0, ) , and the intercept is (0, 1) .

The number e can be approximated by the expression

(1 + 1/x)x for large values of x.

IV. Applications (Pages 186 188) After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the formulas: For n compoundings per year: A = P(1 + r/n)nt For continuous compounding: A = Pert Example 4: Find the amount in an account after 10 years if

$6000 is invested at an interest rate of 7%, (a) compounded monthly. (b) compounded continuously.

(a) $12,057.97 (b) $12,082.52

What you should learn How to recognize, evaluate, and graph exponential functions with base e

Homework Assignment Page(s) Exercises

What you should learn How to use exponential functions to model and solve real-life problems

Section 3.2 Logarithmic Functions and Their Graphs 43 Name ___________________________________________________________ Date ____________


Section 3.2 Logarithmic Functions and Their Graphs Objective: In this lesson you learned how to recognize, evaluate, and

graph logarithmic functions. I. Logarithmic Functions (Pages 192 193) The logarithmic function with base a is the inverse

function of the exponential function xaxf )( .

The logarithmic function with base a is defined as

f(x) = loga x , for x > 0, a > 0, and a 1, if and

only if x = ay. The notation “ loga x ” is read as “ log

base a of x .”

The equation x = ay in exponential form is equivalent to the

equation y = loga x in logarithmic form.

When evaluating logarithms, remember that a logarithm is a(n)

exponent . This means that xalog is the exponent

to which a must be raised to obtain x .

Example 1: Use the definition of logarithmic function to

evaluate 125log 5 . 3 Example 2: Use a calculator to evaluate 300log 10 . 2.477121255

Important Vocabulary Define each term or concept. Common logarithmic function The logarithmic function with base 10. Natural logarithmic function The logarithmic function with base e given by f(x) = ln x, x > 0.

What you should learn How to recognize and evaluate logarithmic functions with base a



Complete the following properties of logarithms:

1) 1log a = 0 2) aalog = 1

3) xa alog = x and xaa log = x

4) If yx aa loglog , then x = y .

Example 3: Solve the equation 1log 7 x for x. x = 7 II. Graphs of Logarithmic Functions (Pages 194 195) For a > 1, is the graph of ( ) logaf x x increasing or decreasing

over its domain? Increasing

For the graph of ( ) logaf x x , a > 1, the domain is

(0, ) , the range is ( , ) , and

the intercept is (1, 0) .

Also, the graph has the y-axis as a vertical

asymptote. The graph of ( ) logaf x x is a reflection of the

graph of ( ) xf x a in the line y = x .

Example 4: Sketch the graph of the function xxf 3log)( .

What you should learn How to graph logarithmic functions with base a

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Section 3.2 Logarithmic Functions and Their Graphs 45


III. The Natural Logarithmic Function (Pages 196 197) Complete the following properties of natural logarithms:

1) 1ln = 0 2) eln = 1

3) xeln = x and xe ln = x

4) If yx lnln , then x = y .

Example 5: Use a calculator to evaluate 10ln . 2.302585093 Example 6: Find the domain of the function )3ln()( xxf . ( 3, ) IV. Applications of Logarithmic Functions (Page 198) Describe a real-life situation in which logarithms are used. Answers will vary. Example 7: A principal P, invested at 6% interest and

compounded continuously, increases to an amount K times the original principal after t years, where t

is given by 06.0

ln Kt . How long will it take the

original investment to double in value? To triple in value?

11.55 years; 18.31 years

What you should learn How to recognize, evaluate, and graph natural logarithmic functions

What you should learn How to use logarithmic functions to model and solve real-life problems



Additional notes


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Section 3.3 Properties of Logarithms 47 Name ___________________________________________________________ Date ____________


Section 3.3 Properties of Logarithms Objective: In this lesson you learned how to rewrite logarithmic functions with

different bases and how to use properties of logarithms to evaluate, rewrite, expand, or condense logarithmic expressions.

I. Change of Base (Page 203) Let a, b, and x be positive real numbers such that a 1 and b 1. The change-of-base formula states that: loga x can be converted to a different base using any of the following formulas: Base b: loga x = (logb x)/(logb a) Base 10: loga x = (log10 x)/(log10 a) Base e: loga x = (ln x)/(ln a) Explain how to use a calculator to evaluate 20log 8 . Using the change-of-base formula, evaluate either (log 20) (log 8) or (ln 20) (ln 8). The results will be the same: 1.4406 II. Properties of Logarithms (Page 204) Let a be a positive number such that a 1; let n be a real number; and let u and v be positive real numbers. Complete the following properties of logarithms: 1. )(log uva = loga u + loga v

2. vu

alog = loga u loga v

3. n

a ulog = n loga u III. Rewriting Logarithmic Expressions (Page 205) To expand a logarithmic expression means to use

the properties of logarithms to rewrite complicated products,

quotients, and exponential forms into simpler sums, differences,

and products .

Example 1: Expand the logarithmic expression 2

ln4xy .

ln x + 4 ln y ln 2

What you should learn How to rewrite logarithms with different bases

What you should learn How to use properties of logarithms to evaluate or rewrite logarithmic expressions

What you should learn How to use properties of logarithms to expand or condense logarithmic expressions



To condense a logarithmic expression means to use the

properties of logarithms to rewrite the expression as the

logarithm of a single quantity .

Example 2: Condense the logarithmic expression

)1log(4log3 xx . log[x3(x 1)4] IV. Applications of Properties of Logarithms (Page 206) One way of finding a model for a set of nonlinear data is to take the natural log of each of the x-values and y-values of the data set. If the points are graphed and fall on a straight line, then the x-values and the y-values are related by the equation: ln y = m ln x , where m is the slope of the

straight line.

Example 3: Find a natural logarithmic equation for the

following data that expresses y as a function of x. ln y = 2 ln x or ln y = ln x2



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y 7.389 54.598 403.429 2980.958

Section 3.4 Solving Exponential and Logarithmic Equations 49 Name ___________________________________________________________ Date ____________


Section 3.4 Solving Exponential and Logarithmic Equations Objective: In this lesson you learned how to solve exponential and

logarithmic equations. I. Introduction (Page 210) State the One-to-One Property for exponential equations. ax = ay if and only if x = y State the One-to-One Property for logarithmic equations. loga x = loga y if and only if x = y State the Inverse Properties for exponential equations and for logarithmic equations. aloga x = x and loga ax = x Describe some strategies for using the One-to-One Properties and the Inverse Properties to solve exponential and logarithmic equations.

Rewrite the original equation in a form that allows the use of the One-to-One Properties of exponential or logarithmic functions.

Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions.

Rewrite a logarithmic equation in exponential form and apply the Inverse Property of exponential functions.

Example 1: (a) Solve 31log 8 x for x.

(b) Solve 04.05 x for x. (a) x = 2 (b) x = 2 II. Solving Exponential Equations (Pages 211 212) Describe how to solve the exponential equation 9010 x algebraically. Take the common logarithm of each side of the equation and then use the Inverse Property to obtain: x = log 90. Then use a calculator to approximate the solution by evaluating log 90 1.954.

What you should learn How to solve simple exponential and logarithmic equations

What you should learn How to solve more complicated exponential equations



Example 2: Solve 5972xe for x. Round to three decimal places.

x 6.190 III. Solving Logarithmic Equations (Pages 213 215) Describe how to solve the logarithmic equation

)8(log)74(log 66 xx algebraically. Use the One-to-One Property for logarithms to write the arguments of each logarithm as equal: (4x 7) = (8 x). Then solve this resulting linear equation by adding 7 to each side, adding x to each side, and then finally dividing both sides by 5. The solution is x = 3. Example 3: Solve 285ln4 x for x. Round to three decimal

places. x 219.327 Describe a method that can be used to approximate the solutions of an exponential or logarithmic equation using a graphing utility. Use a graphing utility to graph the left side of the equation as y1 and the right side of the equation as y2. Use the intersect feature or the zoom and trace features to approximate the intersection point. IV. Applications of Solving Exponential and Logarithmic Equations (Page 216) Example 4: Use the formula for continuous compounding,

rtPeA , to find how long it will take $1500 to triple in value if it is invested at 12% interest, compounded continuously.

t 9.155 years

What you should learn How to solve more complicated logarithmic equations


What you should learn How to use exponential and logarithmic equations to model and solve real-life problems

Section 3.5 Exponential and Logarithmic Models 51 Name ___________________________________________________________ Date ____________


Section 3.5 Exponential and Logarithmic Models Objective: In this lesson you learned how to use exponential growth

models, exponential decay models, Gaussian models, logistic models, and logarithmic models to solve real-life problems.

I. Introduction (Page 221) The exponential growth model is y = aebx, b > 0 .

The exponential decay model is y = ae–bx, b > 0 .

The Gaussian model is y = ae (x b) /c .

The logistic growth model is y = a/(1 + be rx) .

Logarithmic models are y = a + b ln x and

y = a + b log10 x .

II. Exponential Growth and Decay (Pages 222 224) Example 1: Suppose a population is growing according to the

model teP 05.0800 , where t is given in years. (a) What is the initial size of the population? (b) How long will it take this population to

double? (a) 800 (b) 13.86 years To estimate the age of dead organic matter, scientists use the

carbon dating model R = 1/1012 e t/8245 , which

denotes the ratio R of carbon 14 to carbon 12 present at any time

t (in years).

Example 2: The ratio of carbon 14 to carbon 12 in a fossil is

R = 10 16. Find the age of the fossil. Approximately 75,737 years old

What you should learn How to recognize the five most common types of models involving exponential or logarithmic functions

What you should learn How to use exponential growth and decay functions to model and solve real-life problems

Important Vocabulary Define each term or concept. Bell-shaped curve The graph of a Gaussian model. Logistic curve A model for describing populations initially having rapid growth followed by a declining rate of growth. Sigmoidal curve Another name for a logistic growth curve.

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III. Gaussian Models (Page 225) The Gaussian model is commonly used in probability and

statistics to represent populations that are normally

distributed .

On a bell-shaped curve, the average value for a population is

where the maximum y-value of the function occurs.

Example 3: Draw the basic form of the graph of a Gaussian

model. IV. Logistic Growth Models (Page 226) Give an example of a real-life situation that is modeled by a logistic growth model. Answers will vary. One possibility is a bacteria culture that is initially allowed to grow under ideal conditions, and then under less favorable conditions that inhibit growth. Example 4: Draw the basic form of the graph of a logistic

growth model. V. Logarithmic Models (Page 227) Example 5: The number of kitchen widgets y (in millions)

demanded each year is given by the model )1ln(32 xy , where x = 0 represents the year

2000 and x 0. Find the year in which the number of kitchen widgets demanded will be 8.6 million.

In 2008

What you should learn How to use Gaussian functions to model and solve real-life problems

What you should learn How to use logistic growth functions to model and solve real-life problems



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Section 3.6 Nonlinear Models 53 Name ___________________________________________________________ Date ____________


Section 3.6 Nonlinear Models Objective: In this lesson you learned how to fit exponential,

logarithmic, power, and logistic models to sets of data. I. Classifying Scatter Plots (Page 233) When faced with a set of data to be modeled, what is a good first step in selecting which type of model will best fit the data? Making a scatter plot of the data. II. Fitting Nonlinear Models to Data (Pages 234 235) Describe how to use a graphing utility to fit a nonlinear model to data. Answers will vary. For instance, enter the paired data into a graphing utility and graph the data. Use this scatter plot to decide what type of model would fit the data best. Then use the regression feature of the graphing utility to find the appropriate model, either quadratic, exponential, power, or logarithmic. Graph the data and the model in the same viewing window to see whether the model is a good fit to the data. If deciding among several models, compare the coefficients of determination for each model. The model whose r2-value is closest to 1 is the model that best fits the data. Example 2: Find an appropriate model, either logarithmic or

exponential, for the data in the following table. y = 0.8(1.4)x or y = 0.8e0.336x

What you should learn How to classify scatter plots

What you should learn How to use scatter plots and a graphing utility to find models for data and choose the model that best fits a set of data

x 1 3 5 7 9 y 1.120 2.195 4.303 8.433 16.529



III. Modeling With Exponential and Logistic Functions (Pages 236 237) Example 3: Find a logistic model for the data in the following

table. y = 99.88 1 + 19.67e 0.199x Additional notes

What you should learn How to use a graphing utility to find exponential and logistic models for data


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Documents

Chapter 3 Exponential and Logarithmic Functions 329.pdf · graph exponential functions. I. Exponential Functions (Page 180) Polynomial functions and rational functions are examples