EXAMPLE 1 Graph y = b for 0 < b < 1x

Graph y =12

x

SOLUTION

STEP 1 Make a table of values

STEP 2 Plot the points from the table.

STEP 3 Draw, from right to left, a smooth curve that begins just above the x-axis, passes through the plotted points, and moves up to the left.

EXAMPLE 2 Graph y = ab for 0 < b < 1x

Graph the function.

a. Graph y = 214

x

SOLUTION

Plot (0, 2) and .Then,

from right to left, draw a curve that begins just above the x-axis, passes through the two points, and moves up to the left.

1, 12

a.

EXAMPLE 2 Graph y = ab for 0 < b < 1x

b. Graph y = –325

xGraph the function.

SOLUTION

Plot (0, –3) and .

Then,from right to left, draw a curve that begins just below the x-axis, passes through the two points,and moves down to the left.

b. 1, – 65

EXAMPLE 3 Graph y = ab + k for 0 < b < 1x – h

Graph y = 3 –2. State the domain and range.

12

x+1

SOLUTION

Begin by sketching the graph

of y = , which passes

through (0, 3) and . Then

translate the graph left 1 unit and down 2 units .Notice that the translated graph passes through (– 1, 1) and

312

x

32

1,

– 1 2

0,

EXAMPLE 3 Graph y = ab + k for 0 < b < 1x – h

The graph’s asymptote is the line y = –2. The domain is all real numbers, and the range is y > –2.

EXAMPLE 4 Solve a multi-step problem

• Write an exponential decay model giving the snowmobile’s value y (in dollars) after t years. Estimate the value after 3 years.

• Graph the model.

• Use the graph to estimate when the value of the snowmobile will be $2500.

A new snowmobile costs $4200. The value of the snowmobile decreases by 10% each year.

Snowmobiles

EXAMPLE 4 Solve a multi-step problem

The initial amount is a = 4200 and the percent decrease is r = 0.10. So, the exponential decay model is:

Write exponential decay model.

Substitute 4200 for a and 0.10 for r.

Simplify.

y = a(1 – r) t

= 4200(1 – 0.10)t

= 4200(0.90)t

When t = 3, the snowmobile’s value is y = 4200(0.90)3 = $3061.80.

SOLUTION

STEP 1

EXAMPLE 4 Solve a multi-step problem

The graph passes through the points (0, 4200) and (1, 3780).It has the t-axis as an asymptote. Plot a few other points. Then draw a smooth curve through the points.

Using the graph, you can estimate that the value of the snowmobile will be $2500 after about 5 years.

STEP 2

STEP 3

EXAMPLE 1 Simplify natural base expressions

Simplify the expression.

a. e2 e5 = e2 + 5

= e7

b. 12e4

3e3 = e4 – 34

= 4e

(5 )c. e –3x 2 = 52 (e –3x )2

= 25e –6x

= 25e6x

EXAMPLE 2 Evaluate natural base expressions

Use a calculator to evaluate the expression.

a. e4

b. e –0.09

Expression Keystrokes Display

54.59815003

0.9139311853

[ ]ex4

[ ]ex0.09

EXAMPLE 3 Graph natural base functions

Graph the function. State the domain and range.

a. y = 3e 0.25x

SOLUTION

Because a = 3 is positive and r = 0.25 is positive, the function is an exponential growth function. Plot the points (0, 3) and (1, 3.85) and draw the curve.

The domain is all real numbers, and the range is y > 0.

EXAMPLE 3 Graph natural base functions

Graph the function. State the domain and range.

SOLUTION

The domain is all real numbers, and the range is y > 1.

b. y = e –0.75(x – 2) + 1

a = 1 is positive and r = –0.75 is negative, so the function is an exponential decay function. Translate the graph of y = right 2 units and up 1 unit.

e –0.75x

EXAMPLE 4 Solve a multi-step problem

BiologyThe length l (in centimeters) of a tiger shark can be modeled by the function

e –0.178tl = 337 – 276

where t is the shark’s age (in years).

• Graph the model.

• Use the graph to estimate the length of a tiger shark that is 3 years old.

EXAMPLE 4 Solve a multi-step problem

SOLUTION

STEP 1 Graph the model, as shown.

STEP 2 Use the trace feature to determine that l 175 when t = 3.

The length of a 3year-old tiger shark is about 175 centimeters.

ANSWER

EXAMPLE 5 Model continuously compounded interest

A = Pert

SOLUTION

Finance

You deposit $4000 in an account that pays 6% annual interest compounded continuously. What is the balance after 1 year?

Use the formula for continuously compounded interest.

Write formula.

Substitute 4000 for P, 0.06 for r, and 1 for t.= 4000 e0.06(1)

4247.35 Use a calculator.

The balance at the end of 1 year is $4247.35.ANSWER

EXAMPLE 1 Rewrite logarithmic equations

Logarithmic Form Exponential Form

23 = 8a. =2

log 8 3

40 = 1b. 4

log 1 = 0

=c. 12

log 12 1

=d. 1/4

log –14

121 = 12

4=–11

4

EXAMPLE 2 Evaluate logarithms

4loga. 64

b. 5

log 0.2

Evaluate the logarithm.

blogTo help you find the value of y, ask yourself what

power of b gives you y.

SOLUTION

4 to what power gives 64?a. 4

log43 64, so= 3.=64

5 to what power gives 0.2?b. =5–1 0.2, so –1.0.25

log =

EXAMPLE 2 Evaluate logarithms

Evaluate the logarithm.

blogTo help you find the value of y, ask yourself what

power of b gives you y.

SOLUTION

=–31

5 125, so1/5

log 125 =–3.c. to what power gives 125?15

d. 36 to what power gives 6? 361/2 6, so36

log 6= =12

.

d. 36

log 6

c. 1/5

log 125

EXAMPLE 3 Evaluate common and natural logarithms

Expression Keystrokes Display

a. log 8

b. ln 0.3

Check

8

.3

0.903089987

–1.203972804

100.903 8

0.3e –1.204

EXAMPLE 4 Evaluate a logarithmic model

Tornadoes

The wind speed s (in miles per hour) near the center of a tornado can be modeled by

where d is the distance (in miles) that the tornado travels. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the tornado’s center.

93 log d + 65s =

EXAMPLE 4 Evaluate a logarithmic model

SOLUTION

= 93 log 220 + 65

Write function.

93(2.342) + 65

= 282.806

Substitute 220 for d.

Use a calculator.

Simplify.

The wind speed near the tornado’s center was about 283 miles per hour.

ANSWER

93 log d + 65s =

EXAMPLE 5 Use inverse properties

Simplify the expression.

a. 10log4 b. 5

log 25x

SOLUTION

Express 25 as a power with base 5.

a. 10log4 = 4

b. 5

log 25x = (52) x

5log

=5

log 52x

2x=

Power of a power property

blog xb = x

blog bx = x

EXAMPLE 6 Find inverse functions

Find the inverse of the function.

SOLUTION

b.

a. y = 6 x b. y = ln (x + 3)

a.

6log

From the definition of logarithm, the inverse ofy = 6 x is y = x.

Write original function.y = ln (x + 3)Switch x and y.x = ln (y + 3)

Write in exponential form.

Solve for y.

=ex (y + 3)

=ex – 3 y

ANSWER The inverse of y = ln (x + 3) is y = ex – 3.

EXAMPLE 7 Graph logarithmic functions

Graph the function.

SOLUTION

a. y =3

log x

Plot several convenient points, such as (1, 0), (3, 1), and (9, 2). The y-axis is a vertical asymptote.

From left to right, draw a curve that starts just to the right of the y-axis and moves up through the plotted points, as shown below.

EXAMPLE 7 Graph logarithmic functions

Graph the function.

SOLUTION

b. y =1/2

log x

Plot several convenient points, such as (1, 0), (2, –1), (4, –2), and (8, –3). The y-axis is a vertical asymptote.

From left to right, draw a curve that starts just to the right of the y-axis and moves down through the plotted points, as shown below.

EXAMPLE 8 Translate a logarithmic graph

SOLUTION

STEP 1

Graph . State the domain and range.y =2

log (x + 3) + 1

STEP 2

Sketch the graph of the parent function y = x, which passes through (1, 0), (2, 1), and (4, 2).

2log

Translate the parent graph left 3 units and up 1 unit. The translated graph passes through (–2, 1), (–1, 2), and (1, 3). The graph’s asymptote is x = –3. The domain is x > –3, and the range is all real numbers.