Exponential and Logarithmic Functions

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Name______________________________________________

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.

Important Vocabulary

Define each term or concept.

Algebraic functions aaaaaaaaa aa a aaaa aaa aa aaaaaaaaa aa a aaaaaa aaaaaa aa

aaaaa aaaaaaaaaaaa aaaaaaaaaa aaaaaaaaaa aaaaaa aaa aaaaaa Transcendental functions aaaaaaaaa aaaa aaa aaa aaaaaaaaaa

Natural base e aaa aaaaaaaaaa aaaaaa a a aaaaaaaaaaa a a a

Continuous compounding aaaaaaaaaa aaa aaaaaa aa aaaaaaaaaaaa aa aaa aaaaaaaa aaaaaaaa aaaaaaa aaaaaaa aaaaa aaaaa aa aaaaaaaaaa aaaaaaaaaaaa aaaaa aa aaaaa aa aaa aaaaaaa a a aa aaa

I. Exponential Functions (Page 218) _{What you should learn} How to recognize and evaluate exponential functions with base a The exponential function f with base a is denoted by

_________________, where a > 0, a ≠1, and x is any real

number.

Example 1: Use a calculator to evaluate the expression 53/5. aaaaaaaaaaa

II. Graphs of Exponential Functions (Pages 219−221)

What you should learn How to graph

exponential functions and use the One-to-One Property

For a > 1, is the graph of increasing or decreasing over its domain?

x a y=

_____________ a

For a > 1, is the graph of increasing or decreasing over its domain?

x a y= −

_________________ a

For the graph of y=ax or y=a−x, a > 1, the domain is _________ , the range is ______________ , and the

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Example 2: Sketch the graph of the function f(x)= 3−x. y -5 -3 -1 1 3 5 -5 -3 -1 1 3 5x

The graph of the exponential function passes the

aaaaaaaaaa aaaa Test, and therefore, the function is a one-to-one function (and, thus, has an inverse function).

State the One-to-One Property for exponential functions and explain how it may be used to solve simple exponential equations.

aaa a a a aaa a aaa a a a aa aaa aaaa aa a a aa aa aaaaa aa aaaaaaaaaaa aaaaaaaaa aaaaa aaaa aaaa aa aaa aaaaaaaa aaaa aaa aaaa aaaa aaa aaaa aaaaaa aaaaaaaaa aaaaaa aaa aaaaaaaaaa aaaaaaaaa aa aaaaa aaa aaa aaaaaaaaaa

a a

III. The Natural Base e (Page 222) _{What you should learn} How to recognize, evaluate, and graph exponential functions with base e

The natural exponential function is given by the function __________ . In this function, a is the constant and a is the variable.

Example 3: Use a calculator to evaluate the expression e3/5. aaaaaaaaa

Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide

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Section 3.1 • Exponential Functions and Their Graphs 59

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

applications IV. Applications of Exponential Functions (Pages 223−225)

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

For continuous compounding: ______________________

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. aaa aaaaaaaaaa aaa aaaaaaaaaa

Additional notes y -3 -1 1 3 5 -5 -3 -1 1 3 5x y -3 -1 1 3 5 -5 -3 -1 1 3 5x y -3 -1 1 3 5 -5 -3 -1 1 3 5x

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Additional notes y -5 -3 -1 1 3 5 -5 -3 -1 1 3 5x y -5 -3 -1 1 3 5 -5 -3 -1 1 3 5x y -5 -3 -1 1 3 5 -5 -3 -1 1 3 5x Homework Assignment Page(s) Exercises

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Section 3.2 • Logarithmic Functions and Their Graphs 61

Name______________________________________________

Section 3.2 Logarithmic Functions and Their Graphs

Objective: In this lesson you learned how to recognize, evaluate, and graph logarithmic functions.

Important Vocabulary

Common logarithmic function aaa aaaaaaaaaaa aaaaaaaa aaaa aaaa aaa

Natural logarithmic function aaa aaaaaaaaaaa aaaaaaaa aaaa aaaa a aaaaa a aa aa a a aa

What you should learn How to recognize and evaluate logarithmic functions with base a I. Logarithmic Functions (Pages 229−231)

For x > 0, a > 0, and a ≠ 1, where x = ay

, the logarithmic function with base a is defined as

aaaa a aaa a a , which is read as aaaa aaaa a aa aa .

The logarithmic function with base a is the aaaaaaa a

of the exponential function f(x)=ax.

The equation x = ay in exponential form is equivalent to the equation a a aaa a a in logarithmic form.

When evaluating logarithms, remember that a logarithm is a(n) aaaaaaaa . This means that log_a x is the aaaaaaaa a to which a must be raised to obtain a .

Example 1: Use the definition of logarithmic function to evaluate log₅125.

a

Example 2: Use a calculator to evaluate log₁₀300. aaaaaaaaaaa

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Complete the following properties of logarithms:

1) log_a1 = a 2) a = log_a a a

3) log_aax = a a and _alogax = _{a a} 4) If log_a x=log_a y, then a a a .

Example 3: Solve the equation log₇ x=1 for x.

a a a

II. Graphs of Logarithmic Functions (Pages 231−232)

What you should learn How to graph logarithmic functions

To sketch the graph of y=log_a x, you can use the fact that . . . aaa aaaaaa aa aaaaaaa aaaaaaaaa aaa aaaaaaaaaaa aa aaaa aaaaa aa aaa aaaa a a aa

For a > 1, is the graph of increasing or decreasing over its domain?

x y=log_a

aaaaaaaaaa a

For the graph of y=log_a x, a > 1, the domain is

aaa aa , the range is aa aa aa , and the x-intercept is aaa aa .

Also, the graph has aaa aaaaaa as a vertical asymptote.

Example 4: Sketch the graph of the function f(x)=log₃ x.

y -5 -3 -1 1 3 5 -5 -3 -1 1 3 5x

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Section 3.2 • Logarithmic Functions and Their Graphs 63

Name______________________________________________

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

III. The Natural Logarithmic Function (Pages 233−234) The natural logarithm is written without a base; the base is understood to be a .

Complete the following properties of natural logarithms: 1) ln1 = a 2) ln e = a a

3) lnex = a a and elnx = a a

4) If lnx=lny, then a a a .

Example 5: Use a calculator to evaluate ln10. aaaaaaaaaaa

Example 6: Find the domain of the function f(x)=ln(x+3). aaaa aa

IV. Applications of Logarithmic Functions (Page 235) _{What you should learn} How to use logarithmic functions to model and solve real-life

applications Describe a real-life situation in which logarithms are used.

aaaaaaa aaaa aaaaa

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 K

t= . How long will it take the original investment to double in value? To triple in value?

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Additional notes y x y x y x Homework Assignment Page(s) Exercises

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Section 3.3 • Properties of Logarithms 65

Name______________________________________________

Section 3.3 Properties of Logarithms

Objective: In this lesson you learned how to use the change-of-base formula to rewrite and evaluate logarithmic expressions and how to use properties of logarithms to evaluate, rewrite, expand, or condense logarithmic expressions.

I. Change of Base (Page 239) _{What you should learn}

How to use the change-of-base formula to rewrite and evaluate logarithmic expressions Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1.

Use the Change-of-Base Formula to rewrite log_a x using base b: x

a

log = aaaa aaaaaaa aaa a

Explain how to use a calculator to evaluate log₈20.

aaaaa aaa aaaaaaaaaaaaaa aaaaaaaa aaaaaaaa aaaaaa aaaaa aaa a aaaa aa aa aaa aaa a aaa aaa aaa aaaaaaa aaaa aa aaa aaaaa

II. Properties of Logarithms (Page 240)

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_a(uv) = aaa a a aaa a a a

2. v u

a

log = aaa a a aaa a a a

3. log_aun = a aaa a a

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

III. Rewriting Logarithmic Expressions (Page 241) _{What you should learn} How to use properties of logarithms to expand or condense logarithmic expressions

To expand a logarithmic expression means to . . . . aaa aaa aaaaaaaaaa aa aaaaaaaaaa aa aaaaaaa aaa aaaaaaaaaa aa a aaaa aaaaaaaaaaa aaaaaa aaaaaaaa aaaaaaaa aa aaaaaaaaaaa

Example 1: Expand the logarithmic expression 2 ln 4 xy . aa a a a aa a a aa a

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To condense a logarithmic expression means to . . . . aaa aaa aaaaaaaaaa aa aaaaaaaaaa aa aaaaaaa aaa aaaaaaaaaa aa aaa aaaaaaaaa aa a aaaaaa aaaaaaaaa

Example 2: Condense the logarithmic expression . ) 1 log( 4 log 3 x+ x− aaaaa aa a aa aa a

IV. Applications of Properties of Logarithms (Page 242) _{What you should learn} How to use logarithmic functions to model and solve real-life

applications One way of finding a model for a set of nonlinear data is to take

the natural logarithm 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:

aa a a a aa a a, 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.

x 2.718 7.389 20.086 54.598 y 7.389 54.598 403.429 2980.958 aa a a a aa a aa aa a a aa aa y x y x y x Homework Assignment Page(s) Exercises

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Section 3.4 • Exponential and Logarithmic Equations 67

Name______________________________________________

Section 3.4 Exponential and Logarithmic Equations

Objective: In this lesson you learned how to solve exponential and logarithmic equations.

I. Introduction (Page 246) _{What you should learn}

How to solve simple exponential and logarithmic equations State the One-to-One Property for exponential equations.

a a a aa aaa aaaa aa a a aa a

State the One-to-One Property for logarithmic equations.

aaa a a aaa a aa aaa aaaa aa a a aa a

State the Inverse Properties for exponential equations and for logarithmic equations.

aaaaa a a a aaa aaa a a a a

a

Describe how the One-to-One Properties and the Inverse Properties can be used to solve exponential and logarithmic equations.

Example 1: (a) Solve

3 1 log₈ x= for x. (b) Solve 5x =0.04 for x. aaa a a a aaa a a a a

II. Solving Exponential Equations (Pages 247−248)

What you should learn How to solve more complicated exponential equations

Describe how to solve the exponential equation 10x =90. aaaa aaa aaaaaa aaaaaaaaa aaaa aaaa aa aaa aaaaaaaa aaa aaaa aaa aaa aaaaaaa aaaaaaaa aa aaaaaaa a a aaa aaa aaaa aaa a

aaaaaaaaaa aa aaaaaaaaaaa aaa aaaaaaaa aa aaaaaaaaaa aaa aa a aaaaaa

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Example 2: Solve for x. Round to three decimal places. 59 7 2 ₋ ₌ − x e a a aaaaa

What you should learn How to solve more complicated logarithmic equations

III. Solving Logarithmic Equations (Pages 249−250) Describe how to solve the logarithmic equation

. ) 8 ( log ) 7 4 ( log₆ x− = ₆ −x

Example 3: Solve for x. Round to three decimal places. 28 5 ln 4 x= a a aaaaaaa

IV. Applications of Exponential and Logarithmic Equations _{What you should learn} How to use exponential and logarithmic equations to model and solve real-life applications (Pages 251−252)

Example 4: Use the formula for continuous compounding, , to find how long it will take $1500 to triple in value if it is invested at 12% interest, compounded continuously. rt Pe A= a a aaaaa aaaaa Homework Assignment Page(s) Exercises

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Section 3.5 • Exponential and Logarithmic Models 69

Name______________________________________________

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 growth models, and logarithmic models to solve real-life problems.

Important Vocabulary

Bell-shaped curve aaa aaaaa aa a aaaaaaaa aaaaaa

Logistic curve a aaaaa aaa aaaaaaaaaa aaaaaaaaaaa aaaaaaaaa aaaaaa aaaaa aaaaaa aaaaaaaa aa a aaaaaaaaa aaaa aa aaaaaaa

Sigmoidal curve aaaaaaa aaaa aaa a aaaaaaaa aaaaaa aaaaaa

I. Introduction (Page 257) _{What you should learn}

How to recognize the five most common types of models involving exponential and logarithmic functions The exponential growth model is a______________.

The exponential decay model is a______________

The Gaussian model is a__________ a .

aa

The logistic growth model is a______________

Logarithmic models are a______________ and

a______________

II. Exponential Growth and Decay (Pages 258−260)

What you should learn How to use exponential growth and decay functions to model and solve real-life problems Example 1: Suppose a population is growing according to the

model P=800e0.05t, 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?

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To estimate the age of dead organic matter, scientists use the carbon dating model a________________ a, 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.

aaaaaaaaaaaaa aaaaaa aaaaa aaa

III. Gaussian Models (Page 261) _{What you should learn} How to use Gaussian functions to model and solve real-life problems The Gaussian model is commonly used in probability and

statistics to represent populations that are aaaaaaaa a aaaaaaaaaaa .

For a Gaussian model, the average value for a population can be found . . . aaaa aaa aaaaaaaaaaa aaaaa aa aaaaaaaaa aaaaa aaa aaaaaaa aaaaaaa aa aaa aaaaaaaa aaaaaaa aaa aaaaaaa aaaaaaaaaaaaa aa aaa aaaaaaa aaaaaaa aa aaa aaaaaaaa aaaaaaaaaa aaa aaaaaaa aaaaa aa aaa aaaaaaaaaaa aaaaaaaaa

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

y

x

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Section 3.5 • Exponential and Logarithmic Models 71

Name______________________________________________

IV. Logistic Growth Models (Page 262) _{What you should learn} How to use logistic growth functions to model and solve real-life problems

Give an example of a real-life situation that is modeled by a logistic growth model.

aaaaaaa aaaa aaaaa aaa aaaaaaaaaaa aa a aaaaaaaa aaaaaaa aaaa aa aaaaaaaaa aaaaaaa aa aaaa aaaaa aaaaa aaaaaaaaaaa aaa aaaa aaaaa aaaa aaaaaaaaa aaaaaaaaaa aaaa aaaaaaa aaaaaaaa Example 4: Draw the basic form of the graph of a logistic

growth model.

y

x

V. Logarithmic Models (Page 263) _{What you should learn} How to use logarithmic functions to model and solve real-life problems Example 5: The number of kitchen widgets y (in millions)

demanded each year is given by the model

, 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.

) 1 ln( 3 2+ + = x y aa aaaa

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Additional notes y x y x y x Homework Assignment Page(s) Exercises