Copyright © Cengage Learning. All rights reserved. 11 Exponential and Logarithmic Functions

Copyright © Cengage Learning. All rights reserved.

11Exponential andLogarithmic Functions

11.1 EXPONENTIAL FUNCTIONS AND THEIR GRAPHS

Copyright © Cengage Learning. All rights reserved.

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• Recognize and evaluate exponential functions with base a.

• Graph exponential functions and use the One-to-One Property.

• Recognize, evaluate, and graph exponential functions with base e.

• Use exponential functions to model and solve real-life problems.

What You Should Learn

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Exponential Functions

5

Exponential Functions

So far, this text has dealt mainly with algebraic functions, which include polynomial functions and rational functions.

In this chapter, you will study two types of nonalgebraic functions–exponential functions and logarithmic functions. These functions are examples of transcendental functions.

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Exponential Functions

The base a = 1 is excluded because it yields f (x) = 1x = 1. This is a constant function, not an exponential function.

You have evaluated ax for integer and rational values of x.

For example, you know that 43 = 64 and 41/2 = 2.

However, to evaluate 4x for any real number x, you need to interpret forms with irrational exponents.

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Exponential Functions

For the purposes of this text, it is sufficient to think of

(where 1.41421356)

as the number that has the successively closer approximations

a1.4, a1.41, a1.414, a1.4142, a1.41421, . . . .

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Example 1 – Evaluating Exponential Functions

Use a calculator to evaluate each function at the indicated value of x.

Function Value

a. f (x) = 2x x = –3.1

b. f (x) = 2–x x =

c. f (x) = 0.6x x =

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Example 1 – Solution

Function Value Graphing Calculator Keystrokes Display

a. f (–3.1) = 2–3.1 0.1166291

b. f () = 2– 0.1133147

c. f = 0.63/2 0.4647580

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Graphs of Exponential Functions

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Example 2 – Graphs of y = ax

In the same coordinate plane, sketch the graph of each function.

a. f (x) = 2x

b. g(x) = 4x

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Example 2 – Solution

The table below lists some values for each function, and Figure 3.1 shows the graphs of the two functions.

Note that both graphs are increasing. Moreover, the graph of g(x) = 4x is increasing more rapidly than the graph off (x) = 2x.

Figure 3.1

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Graphs of Exponential Functions

The basic characteristics of exponential functions y = ax and

y = a–x are summarized in Figures 3.3 and 3.4.

Graph of y = ax, a > 1

• Domain: ( , )

• Range: (0, )

• y-intercept: (0, 1)

• Increasing

• x-axis is a horizontal asymptote (ax→ 0, as x→ ).

• Continuous

Figure 3.3

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Graphs of Exponential Functions

Graph of y = a–x, a > 1

• Domain: ( , )

• Range: (0, )

• y-intercept: (0, 1)

• Decreasing

• x-axis is a horizontal asymptote (a–x → 0, as x→ ).

• Continuous

From Figures 3.3 and 3.4, you can see that the graph of an exponential function is always increasing or always decreasing.

Figure 3.4

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Graphs of Exponential Functions

As a result, the graphs pass the Horizontal Line Test, and therefore the functions are one-to-one functions.

You can use the following One-to-One Property to solve simple exponential equations.

For a > 0 and a ≠ 1, ax = ay if and only if x = y.

One-to-One Property

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The Natural Base e

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The Natural Base e

In many applications, the most convenient choice for a base is the irrational number

e 2.718281828 . . . .

This number is called the natural base.

The function given by f (x) = ex is

called the natural exponential

function. Its graph is shown in

Figure 3.9.

Figure 3.9

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The Natural Base e

Be sure you see that for the exponential function

f (x) = ex, e is the constant 2.718281828 . . . , whereas x

is the variable.

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Example 6 – Evaluating the Natural Exponential Function

Use a calculator to evaluate the function given by

f (x) = ex at each indicated value of x.

a. x = –2

b. x = –1

c. x = 0.25

d. x = –0.3

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Example 6 – Solution

Function Value Graphing Calculator Keystrokes Display

a. f (–2) = e–2 0.1353353

b. f (–1) = e–1 0.3678794

c. f (0.25) = e0.25 1.2840254

d. f (–0.3) = e–0.3 0.7408182

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Definition of the Exponential Function

The exponential function f with base b is defined byf (x) = bx or y = bx

Where b is a positive constant other than and x is any real number.

The exponential function f with base b is defined byf (x) = bx or y = bx

Where b is a positive constant other than and x is any real number./

Here are some examples of exponential functions.f (x) = 2x g(x) = 10x h(x) = 3x+1

Base is 2. Base is 10. Base is 3.

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Text ExampleThe exponential function f (x) = 13.49(0.967)x – 1 describes the number of O-rings expected to fail, when the temperature is x°F. On the morning the Challenger was launched, the temperature was 31°F, colder than any previous experience. Find the number of O-rings expected to fail at this temperature.

Solution Because the temperature was 31°F, substitute 31 for x and evaluate the function at 31.

f (x) = 13.49(0.967)x – 1 This is the given function.

f (31) = 13.49(0.967)31 – 1 Substitute 31 for x.

f (31) = 13.49(0.967)31 – 1=3.77

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Characteristics of Exponential Functions1. The domain of f (x) = bx consists of all real numbers. The range of f

(x) = bx consists of all positive real numbers.2. The graphs of all exponential functions pass through the point (0,

1) because f (0) = b0 = 1.3. If b > 1, f (x) = bx has a graph that goes up to the right and is an

increasing function.4. If 0 < b < 1, f (x) = bx has a graph that goes down to the right and is

a decreasing function.5. f (x) = bx is a one-to-one function and has an inverse that is a

function.6. The graph of f (x) = bx approaches but does not cross the x-axis. The

x-axis is a horizontal asymptote.

1. The domain of f (x) = bx consists of all real numbers. The range of f (x) = bx consists of all positive real numbers.

2. The graphs of all exponential functions pass through the point (0, 1) because f (0) = b0 = 1.

3. If b > 1, f (x) = bx has a graph that goes up to the right and is an increasing function.

4. If 0 < b < 1, f (x) = bx has a graph that goes down to the right and is a decreasing function.

5. f (x) = bx is a one-to-one function and has an inverse that is a function.

6. The graph of f (x) = bx approaches but does not cross the x-axis. The x-axis is a horizontal asymptote.

f (x) = bx

b > 1 f (x) = bx

0 < b < 1

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Transformations Involving Exponential Functions

• Shifts the graph of f (x) = bx upward c units if c > 0.• Shifts the graph of f (x) = bx downward c units if c < 0.

g(x) = -bx + cVertical translation

• Reflects the graph of f (x) = bx about the x-axis.• Reflects the graph of f (x) = bx about the y-axis.

g(x) = -bx

g(x) = b-x

Reflecting

Multiplying y-coordintates of f (x) = bx by c,• Stretches the graph of f (x) = bx if c > 1.• Shrinks the graph of f (x) = bx if 0 < c < 1.

g(x) = c bxVertical stretching or shrinking

• Shifts the graph of f (x) = bx to the left c units if c > 0.• Shifts the graph of f (x) = bx to the right c units if c < 0.

g(x) = bx+cHorizontal translation

DescriptionEquationTransformation

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Text Example

Use the graph of f (x) = 3x to obtain the graph of g(x) = 3 x+1.

Solution Examine the table below. Note that the function g(x) = 3x+1 has the general form g(x) = bx+c, where c = 1. Because c > 0, we graph g(x) = 3 x+1 by shifting the graph of f (x) = 3x one unit to the left. We construct a table showing some of the coordinates for f and g to build their graphs.

f (x) = 3xg(x) = 3x+1

(0, 1)(-1, 1)

1 2 3 4 5 6-5 -4 -3 -2 -1

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Problems

Sketch a graph using transformation of the following:

1.

2.

3.

Recall the order of shifting: horizontal, reflection (horz., vert.), vertical.

( ) 2 3xf x

( ) 2 1xf x

1( ) 4 1xf x

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The Natural Base e

An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72. More accurately,

The number e is called the natural base. The function f (x) = ex is called the natural exponential function.

2.71828...e

-1

f (x) = ex

f (x) = 2x

f (x) = 3x

(0, 1)

(1, 2)

1

2

3

4

(1, e)

(1, 3)

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Formulas for Compound Interest

After t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas:

1. For n compoundings per year:

2. For continuous compounding: A = Pert.

1rt

rA P

n

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Example: Choosing Between Investments

You want to invest $8000 for 6 years, and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment?

Solution The better investment is the one with the greater balance in the account after 6 years. Let’s begin with the account with monthly compounding. We use the compound interest model with P = 8000, r = 7% = 0.07, n = 12 (monthly compounding, means 12 compoundings per year), and t = 6.

The balance in this account after 6 years is $12,160.84.

12*60.071 8000 1 12,160.8412

ntrA Pn

moremore

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Example: Choosing Between Investments

You want to invest $8000 for 6 years, and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment?

Solution For the second investment option, we use the model for continuous compounding with P = 8000, r = 6.85% = 0.0685, and t = 6.

The balance in this account after 6 years is $12,066.60, slightly less than the previous amount. Thus, the better investment is the 7% monthly compounding option.

0.0685(6)8000 12,066.60rtA Pe e

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Example

Use A= Pert to solve the following problem: Find the accumulated value of an investment of $2000 for 8 years at an interest rate of 7% if the money is compounded continuously

Solution:A= Pert

A = 2000e(.07)(8)

A = 2000 e(.56)

A = 2000 * 1.75A = $3500

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LOGARITMIC FUNCTION

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For x 0 and 0 a 1, y = loga x if and only if x = a y.

The function given by f (x) = loga x is called

the logarithmic function with base a.

Every logarithmic equation has an equivalent exponential form: y = loga x is equivalent to x = a y

A logarithmic function is the inverse function of an exponential function.

Exponential function: y = ax

Logarithmic function: y = logax is equivalent to x = ay

A logarithm is an exponent!

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y = log2( )2

1 = 2 y

2

1

Examples: Write the equivalent exponential equation and solve for y.

1 = 5 yy = log51

16 = 4y y = log416

16 = 2yy = log216

SolutionEquivalent Exponential

Equation

Logarithmic Equation

16 = 24 y = 4

2

1= 2-1 y = –1

16 = 42 y = 2

1 = 50 y = 0

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log10 –4 LOG –4 ENTER ERROR

no power of 10 gives a negative number

The base 10 logarithm function f (x) = log10 x is called the

common logarithm function.

The LOG key on a calculator is used to obtain common logarithms.

Examples: Calculate the values using a calculator.

log10 100

log10 5

Function Value Keystrokes Display

LOG 100 ENTER 2

LOG 5 ENTER 0.69897005

2log10( ) – 0.3979400LOG ( 2 5 ) ENTER

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Examples: Solve for x: log6 6 = x

log6 6 = 1 property 2 x = 1

Simplify: log3 35

log3 35 = 5 property 3

Simplify: 7log79

7log79 = 9 property 3

Properties of Logarithms

1. loga 1 = 0 since a0 = 1.

2. loga a = 1 since a1 = a.

4. If loga x = loga y, then x = y. one-to-one property

3. loga ax = x and alogax = x inverse property

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x

y

Graph f (x) = log2 x

Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x.

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42

21

10

–1

–2

2xx

4

1

2

1

y = log2 x

y = xy = 2x

(1, 0)

x-intercept

horizontal asymptote y = 0

vertical asymptote x = 0

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Example: Graph the common logarithm function f(x) = log10 x.

by calculator

1

10

10.6020.3010–1–2f(x) = log10 x

10421x 1

100

y

x

5

–5

f(x) = log10 x

x = 0 vertical asymptote

(0, 1) x-intercept

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The graphs of logarithmic functions are similar for different values of a.

f(x) = loga x (a 1)

3. x-intercept (1, 0)

5. increasing

6. continuous

7. one-to-one

8. reflection of y = a x in y = x

1. domain ),0( 2. range ),(

4. vertical asymptote

)(0 as 0 xfxx

Graph of f (x) = loga x (a 1)

x

yy = x

y = log2 x

y = a x

domain

range

y-axisverticalasymptote

x-intercept(1, 0)

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The function defined by f(x) = loge x = ln x

is called the natural logarithm function.

Use a calculator to evaluate: ln 3, ln –2, ln 100

ln 3

ln –2

ln 100

Function Value Keystrokes Display

LN 3 ENTER 1.0986122

ERRORLN –2 ENTER

LN 100 ENTER 4.6051701

y = ln x

(x 0, e 2.718281)

y

x

5

–5

y = ln x is equivalent to e y = x

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Properties of Natural Logarithms

1. ln 1 = 0 since e0 = 1.

2. ln e = 1 since e1 = e.

3. ln ex = x and eln x = x inverse property

4. If ln x = ln y, then x = y. one-to-one property

Examples: Simplify each expression.

2

1lne

2ln 2 e inverse property

20lne 20 inverse property

eln3 3)1(3 property 2

00 1ln property 1

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Example: The formula (t in years) is used to estimate the age of

organic material. The ratio of carbon 14 to carbon 12

in a piece of charcoal found at an archaeological dig is . How old is

it?

82231210

1 t

eR

To the nearest thousand years the charcoal is 57,000 years old.

original equation15

822312 10

1

10

1

t

e

multiply both sides by 1012

1000

18223 t

e

take the natural log of both sides1000

1lnln 8223

t

e

inverse property1000

1ln

8223

t

56796907.6 82231000

1ln 8223

t

1510

1R