Algebra 2

6 Chapter Test athXM

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Do you know HOW? Find each product or quotient.
Simplify each radical expression. Use absolute value
symbols when needed. 21. 15114 52 22. 2x3
15 x2
1. 254x3y5
For each pair of functions, find (g ∘ f )(x) and ( f ∘ g)(x).
2. 23 - 0.027 23. f (x) = x2 - 2, g (x) = 4x + 1
3. 25 - 64x14y20 24. f (x) = 2x2 + x - 7, g (x) = -3x - 1

Simplify each expression. Rationalize all Find the inverse of each function. Is the inverse a
function?
#denominators.
25. f (x) = 1x + 322 + 1
4. 27x3 114x
26. f (x) = 12x + 1
5. 1 - 13x 27. g (x) = 3x3 - 4
16x 28. f (x) = 41x

6. 148 + 2127 + 5112

7. 13 + 215211 - 1202

8. 417xz + 217xz Rewrite each function to make it easy to graph using
transformations. Describe the graph.
9. 175 12
- 12 29. y = 116x + 80 - 1 30. y = 19x + 3

Simplify each expression. #1 1 Graph. Find the domain and range of each function.
31. y = 21x + 3
10. (125)- 2 11. x6 x3 32. y = - 12x + 3
3 13. 28x5 - 218x5 33. y = 1x + 3 - 4

2

8x9y3 3
27x2y12≤
12. ¢

Solve each equation. Check for extraneous solutions. Do you UNDERSTAND?

14. 1x - 3 = x - 5 15. 1x + 4 = 13x 34. Writing  ​Explain why -108 has no real 6th roots.
17. 1x + 3 - 1 = x
3 35. Open-Ended  ​Write a relation that is not a function,
but whose inverse is a function.
16. 2(x - 1)4 = 16
36. Measurement  ​The time t in seconds for a swinging
Let f (x) = x − 2 and g (x) = x2 − 3x + 2. Perform pendulum to complete one full cycle is given by
each function operation and then find the domain. the function t = 0.21l, where l is the length of the
pendulum in centimeters. To the nearest tenth, how
# 18. -2g(x) + f (x) long is a full cycle if the pendulum is 10 cm long?
20 cm long? How long, in centimeters, is a pendulum
19. -f (x) g (x) that takes 2 seconds for one full cycle?

20. g(x)
f (x)

Chapter 6  Chapter Test 427

6 Common Core Cumulative ASSESSMENT
Standards Review

Some problems require you to What is the inverse of the function TIP 2
find the inverse of a function. y = x2 + 3?
After you interchange
TIP 1 y = x - 3 x and y, solve for y.

To find the inverse of a y = { 1x - 3 Think It Through
function, interchange x y = { 2x2 + 3
and y. y = (x - 3)2 y = x2 + 3
x = y2 + 3
x - 3 = y2
{ 1x - 3 = y
y = { 1x - 3
The correct answer is B.

LVVeooscsacoabnubluarlayry Review Selected Response

As you solve test items, you must understand the Read each question. Then write the letter of the correct
meanings of mathematical terms. Match each term answer on your paper.
with its mathematical meaning.
1. Find all the roots of 2x4 + x3 - 8x2 - 4x = 0.
A . radicand I.  the combination of two functions x = -2, x = -0.5, x = 0, x = 2
B. index such that the output from the first
C . cfuonmcptioosnite becomes the input for the second x = -2, x = -0.5, x = 2
D . ifnuvnecrtsioens
E . fruadnicctaiol n II. t he degree of a root in a radical x = -2, x = 0.5, x = 0, x = 2
expression
x = -2, x = 0.5, x = 2
III.  the number under the radical
sign in a radical expression 2. Solve the equation ax2 + bx + c = 0 for b.

IV. a function that can be written in b = - cx - ax2 b = - (cx - ax2)
the form f (x) = a1n x - h + k
b = -c - ax2 b = - (c - ax2)
x x

V. t he range of one function is the 3. Use the sum of cubes formula to factor x3 + 64.
domain of the other and vice (x + 4)(x2 - 4x + 4)
versa
(x + 4)(x2 + 4x + 4)

(x + 4)(x2 - 4x + 16)

(x + 4)(x2 + 4x + 16)

428 Chapter 6  Common Core Cumulative Standards Review

4. The time it takes to copy pages varies directly with 8. Solve 7x2 + 196 = 0 for x.
{4i17
the number of pages being copied. The copier at your {417
{2i17
office can copy 21 color pages per minute and 40 black {217

and white pages per minute. Approximately how long 9. Which inequality is modeled by the graph?

will it take to copy 60 color pages and 35 black and

white pages?

0.9 minute 2.9 minutes

2.5 minutes 3.7 minutes

5. Which equation is modeled by the graph? 12 3 4 5 6 7

6y k + 1 … 7

4 0k + 10 … 7

k - 4 … 3

0k - 40 … 3

x 1 0. What is the solution of the system? 4x + 2y = 4
e 6x + 2y = 8
4 2 O

y = 0 2x - 3 0 y = 0 2x + 3 0 ( -2, 2)

y = 20x - 30 y = 20x + 30 (2, -2)

6. What are the vertex and axis of symmetry for the given (1, 2)
parabola?
( -1, 2)

4y 11. A photographer is promoting three photo specials.
How much does it cost for each type of print?

2 One 5x7 Two 5x7s
Three 3x5s Five 3x5s
x One wallet sheet One wallet sheet
O 2468

2

4 Four 5x7s
Four 3x5s
Two wallet sheets

( -4, 5), y = 5 5 * 7 costs $7, 3 * 5 costs $5, Wallet costs $3
( -4, 5), x = 5
(5, -4), y = -4 5 * 7 costs $11, 3 * 5 costs $7, Wallet costs $3
(5, -4), x = 5
7. What is the product of 13 3 and 15 3? 5 * 7 costs $12, 3 * 5 costs $11, Wallet costs $7
18 3
18 9 5 * 7 costs $7, 3 * 5 costs $5, Wallet costs $5
215 38
28 315 1 2. What is an equivalent form of 2 5 2i ?
+

45i 5 + 5i
4

140 - 41i0i 5 - 5i
- 4

Chapter 6  Common Core Cumulative Standards Review 429

Constructed Response 2 3. The total cost (in cents) of x + 2 markers is
x3 + 5x2 + 2x - 8.
13. Let g (x) = x - 3 and h (x) = x2 + 6. What is h(1) * g(1)?
14. A laptop comes without any programs installed on it. a. Write an expression that models the cost of each
marker.
Each program costs $20 and the laptop you want costs
$319. What is the greatest number of programs you can b. If you buy 7 markers, how much would each
buy if you want to spend at most $500 for the laptop? marker cost?

15. You are building an entertainment center with shelves 24. You are given that f (x) = x2 + 4 and g(x) = 3x - 1.
that are x in. deep by x in. long. The height of the a. What are the domain and range of f (x) and g(x)?
unit will be twice the depth. If the volume of the unit b. Find f (x) + g(x).
will be 8,192 in.3, what is the height, in inches, of the c. What is the domain of your answer to part (b)?
entertainment center?
25. Two friends went shopping together. One friend
16. What is the solution of the equation bought 2 hats and 1 shirt and spent $70, while the
x2 - 24x + 144 = 0? other friend bought 1 hat and 3 shirts and spent $85.
Use a graph to determine the costs of each shirt and
1 7. What is the number of real roots of the equation hat.
2x2 + 3x = - 4?
26. A student found that a cubic function has zeros 16
18. What is the quotient 23184xx64yy182? and 1 - 2i with a leading coefficient of 3. What is the
constant term of this polynomial function with real
1 9. What is the solution of 4 + 13x + 5 = 7? coefficients?

20. All 385 tickets for a high-school play sold in 10 days. 27. Describe the graph of the polynomial function
The ticket receipts totaled $1960. If the cost of a child’s f (x) = -x6 + 3x5 + 4x - 10. What is its end
ticket was $4 and the cost of an adult’s ticket was $6, behavior?
how many adult tickets were sold?
Decide whether the following statements are always,
21. What is the x-value of the x-intercept of the graph of sometimes, or never true.
f (x) = x2 + 4x + 4?
2 8. If n is a real number, then 0n = 0.
2 2. The graph shows a transformation of f (x) = x2. What 2 9. If a and b are rational numbers, then the product of
is an equation of the graph? Explain your answer by
using translations of the parent quadratic function. (a + 1b) and its conjugate is a rational number.

y Extended Response
g(x) 8
3 0. An online music store is having a promotion.
6 Customers receive a $5 rebate if they buy any regular
priced CD at $13 each. They can also receive 15% off if
4 they register as a store member.

2 x a. What functions model the two discounts?
Ϫ4 Ϫ2 O 2 b. In which order should the discounts be applied

for the customer to receive the greatest discount?
c. Use your answer from part (b) to determine the

amount a customer will save if she buys 5 CDs.

430 Chapter 6  Common Core Cumulative Standards Review

Get Ready! CHAPTER

Lesson 1-3 Evaluating Expressions 7

Evaluate each expression for x = −2, 0, and 2.

( ) 1. 10x+1 2. 3 x 3. - 5x-2 4. - (3)0.5x
2

Lesson 2-5 Using Linear Models

Draw a scatter plot and find the line of best fit for each set of data.

5. (0, 2), (1, 4), (2, 6.5), (3, 8.5), (4, 10), (5, 12), (6, 14)

6. (3, 100), (5, 150), (7, 195), (9, 244), (11, 296), (13, 346), (15, 396)

Lessons 4-1 Graphing Transformations

and 5-9 Identify the parent function of each equation. Graph each equation as a
transformation of its parent function.

7. y = (x + 5)2 - 3 8. y = - 2 (x - 6)3

Lesson 6-4 Simplifying Rational Exponents

Simplify each expression.

9. 1x 51210 10. 1 - 8x 3234

Lesson 6-7 Finding Inverses

Find the inverse of each function. Is the inverse a function?

11. y = 10 - 2x 2 12. y = (x + 4)3 - 1

Looking Ahead Vocabulary 431

13. In advertising, the decay factor describes how an advertisement loses its
effectiveness over time. In math, would you expect a decay factor to increase or
decrease the value of y as x increases?

14. There are many different kinds of growth patterns. Patterns that increase by
a constant rate are linear. Patterns that grow exponentially increase by an
ever-increasing rate. If your allowance doubles each week, does that represent
linear growth or exponential growth?

15. The word asymptote comes from a Greek word meaning “not falling together.”
When looking at the end behavior of a function, do you expect the graph to
intersect its asymptote?

Chapter 7  Exponential and Logarithmic Functions

CHAPTER Exponential and

7 Logarithmic Functions

Download videos VIDEO Chapter Preview 1 Modeling
connecting math Essential Question  How do you model a
to your world.. 7-1 Exploring Exponential Models quantity that changes regularly over time
7-2 Properties of Exponential Functions by the same percentage?
Interactive! ICYNAM 7-3 Logarithmic Functions as Inverses
Vary numbers, ACT I V I TI 7-4 Properties of Logarithms 2 Equivalence
graphs, and figures D 7-5 Exponential and Logarithmic Equations Essential Question  How are exponents
to explore math ES 7-6 Natural Logarithms and logarithms related?
concepts..
3 Function
The online Essential Question  How are exponential
Solve It will get functions and logarithmic functions
you in gear for related?
each lesson.

Math definitions VOC ABUL ARY Vocabulary DOMAINS
in English and • Linear and Exponential Models
Spanish English/Spanish Vocabulary Audio Online: • Creating Equations that Describe Numbers
English Spanish • Interpreting Functions
asymptote, p. 435 asíntota
Online access O NLINE Change of Base Formula, fórmula de cambio de base
to stepped-out RKME WOp. 464
problems aligned HO common logarithm, p. 453 logaritmo común
to Common Core exponential equation, p. 469 ecuación exponencial
Get and view exponential function, p. 434 función exponencial
your assignments exponential growth, p. 435 incremento exponencial
online. logarithm, p. 451 logaritmo
logarithmic equation, p. 471 ecuación logarítmica
Extra practice logarithmic function, p. 454 función logarítmica
and review natural logarithmic function, función logarítmica natural
online p. 478

Virtual NerdTM
tutorials with
built-in support

PERFORMANCE TASK

Common Core Performance Task

Apparent Magnitudes of Stars

Astronomers refer to the brightness of a star as its apparent magnitude. Apparent
magnitude is measured on a decreasing scale, meaning that brighter stars have
lower apparent magnitudes. For example, Polaris (the North Star) is one of the
brighter stars in the night sky and has an apparent magnitude of 1.97, while stars
that can barely be seen with the unaided eye have apparent magnitudes of about
6.5. The Sun has an apparent magnitude of -26.74.
Apparent magnitude does not indicate how brightly a star burns. Many stars
burn brighter than our Sun, but they appear faint and dim because of their great
distance from us.

In the apparent magnitude scale, a decrease of 1 unit corresponds to an increase
in brightness by a factor of 15 100. For example, a star of magnitude 3 and a star
of magnitude 1 are separated by 2 units on the apparent magnitude scale, so the
star of magnitude 1 is (15 100)2 times (or about 6.3 times) as bright as the star of
magnitude 3.

Task Description

Sirius, the brightest star in the night sky, is about 24 times as bright as Polaris.
What is the apparent magnitude of Sirius?

Connecting the Task to the Math Practices MATHEMATICAL

As you complete the task, you’ll apply several Standards for Mathematical PRACTICES
Practice.

• You’ll make sense of the information provided and write an exponential
function that models the problem situation. (MP 1, MP 2)

• You’ll solve an equation algebraically and by using a graph. (MP 5)

Chapter 7  Exponential and Logarithmic Functions 433

7-1 Exploring Exponential MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Models
FM-IAF.FCS..79e1 2G.Fra-IpFh.3e.x7peo nGenratipahl .e.x.pfounectnitoinasl,.s.h.ofwunincgtions,
Objective To model exponential growth and decay isnhtoewrcienpgtsinatnedrcenpdtsbaenhdaveinodr .b.e.hAalvsioorA.-.S.SAEl.sAo.1b,
AM-ACFESD..9A1.2,.AF-SIFS.CE..81.1b, MAFS.912.A-CED.1.2,
MMPAF1S,.M91P2.2F,-MIF.P3.38, MP 4, MP 5
MP 1, MP 2, MP 3, MP 4, MP 5

Hmmm, I wonder if You are to move the stack of 5 rings to another
there is a shortcut? post. Here are the rules.

MATHEMATICAL • A move consists of taking the top ring from
one post and placing it onto another post.
PRACTICES
• You can move only one ring at a time.

• Do not place a ring on top of a smaller ring.

What is the fewest number of moves needed?
How many moves are needed for 10 rings? 20 rings? Explain.

Lesson The number of moves needed for additional rings in the Solve It suggests a pattern that
approximates repeated multiplication.
Vocabulary
• exponential Essential Understanding  Y​ ou can represent repeated multiplication with a
function of the form y = abx where b is a positive number other than 1.
function An exponential function is a function with the general form y = abx, a ≠ 0, with
• exponential b 7 0, and b ≠ 1. In an exponential function, the base b is a constant. The exponent x
is the independent variable with domain the set of real numbers.
growth
• exponential

decay
• asymptote
• growth factor
• decay factor

Problem 1 Graphing an Exponential Function

How does making a What is the graph of y = 2x? Step 2 Plot and connect the points.
table help you sketch Step 1 Make a table of values.
the graph?
x 2x y x 2x y 8y
The table shows 0 20 1 6
Ϫ4 2Ϫ4 1 ϭ 0.0625 1 21 2 4
coordinates of several 16 2 22 4 2
3 23 8
points on the graph. Ϫ3 2Ϫ3 1 ϭ 0.125 Ϫ4 Ϫ2 O 2
8

Ϫ2 2Ϫ2 1 ϭ 0.25
4

Ϫ1 2Ϫ1 1 ϭ 0.5 x
2 4

434 Chapter 7  Exponential and Logarithmic Functions

Got It? 1. What is the graph of each function? c. y = 2(3)x

( ) a. y = 4x b. y = 31 x
d. Reasoning ​What generalizations can you make about the domain,

range, and y-intercepts of these functions?

Two types of exponential behavior are y
exponential growth and exponential decay.

For exponential growth, as the value of x

increases, the value of y increases. For Exponential Exponential
exponential decay, as the value of x increases, Decay Growth
the value of y decreases, approaching zero.
x
The exponential functions shown here are asymptotic
to the x-axis. An asymptote is a line that a graph O

approaches as x or y increases in absolute value. The x-axis is an asymptote.

Concept Summary  Exponential Functions

For the function y = abx,
• if a 7 0 and b 7 1, the function represents exponential growth.
• if a 7 0 and 0 6 b 6 1, the function represents exponential decay.
In either case, the y-intercept is (0, a), the domain is all real numbers,
the asymptote is y = 0, and the range is y 7 0.

Problem 2 Identifying Exponential Growth and Decay

Identify each function or situation as an example of exponential growth or decay.
What is the y-intercept?

A y = 12(0.95)x B y = 0.25(2)x

Since 0 6 b 6 1, the function Since b 7 1, the function

What quantity does represents exponential decay. represents exponential growth.
the y-intercept The y-intercept is (0, a) = (0, 12). The y-intercept is (0, a) = (0, 0.25).

rT eheprye-isnetenrtc?ept is the C You put $1000 into a college savings account for four years. The account pays
amount of money at 5% interest annually.

t = 0, which is the The amount of money in the bank grows by 5% annually. It represents exponential
initial investment.
growth. The y-intercept is 1000, which is the dollar value of the initial investment.

Got It? 2. Identify each function or situation as an example of exponential growth or
decay. What is the y-intercept?

a. y = 314x2 b. y = 1110.75x2

c. You put $2000 into a college savings account for four years. The account
pays 6% interest annually.

Lesson 7-1  Exploring Exponential Models 435

For exponential growth y = abx, with b 7 1, the value b is the growth factor.
A quantity that exhibits exponential growth increases by a constant percentage each
time period. The percentage increase r, written as a decimal, is the rate of increase or
growth rate. For exponential growth, b = 1 + r.

For exponential decay, 0 6 b 6 1 and b is the decay factor. The quantity decreases by
a constant percentage each time period. The percentage decrease, r, is the rate of decay.
Usually a rate of decay is expressed as a negative quantity, so b = 1 + r.

Key Concept  Exponential Growth and Decay
You can model exponential growth or decay with this function.

Amount after t Rate of growth (r Ͼ 0)
time periods or decay (r Ͻ 0)

A(t) ϭ a(1 ϩ r)t

Initial amount Number of
time periods

For growth or decay to be exponential, a quantity changes by a fixed percentage each
time period.

Problem 3 Modeling Exponential Growth

What is the growth You invested $1000 in a savings account at the end of 6th grade. The account pays 5%
rate r? annual interest. How much money will be in the account after six years?
It is the annual interest
rate, written as a Step 1 Determine if an exponential function is a reasonable model.
decimal: 5, = 0.05.
The money grows at a fixed rate of 5% per year. An exponential model is
appropriate.

Step 2 Define the variables and determine the model.

Let t = the number of years since the money was invested.
Let A (t) = the amount in the account after each year.
A reasonable model is A (t) = a(1 + r)t.
Step 3 Use the model to solve the problem.

A (6) = 1000(1 + 0.05)6 Substitute a = 1000, r = 0.05, and t = 6.
= 1000(1.05)6 Simplify.
≈ +1340.10

The account contains $1340.10 after six years.

Got It? 3. Suppose you invest $500 in a savings account that pays 3.5% annual interest.
How much will be in the account after five years?

436 Chapter 7  Exponential and Logarithmic Functions

Problem 4 Using Exponential Growth

How can you make Suppose you invest $1000 in a savings account that pays 5% annual interest. If you
a table to solve make no additional deposits or withdrawals, how many years will it take for the
this problem? account to grow to at least $1500?
Define the variables,
write an equation and Define the variables. Let t = the number of years.
enter it into a graphing Let A (t) = the amount in the account
calculator. Then you can Determine the model. after t years.
inspect a table to find
the solution. Make a table using A (t) = 1000(1 + 0.05)t
the table feature on a = 1000(1.05)t
graphing calculator. Find
the input when the output
is 1500. X Y1

The account pays interest 4 1215.5
only once a year. The 5 1276.3
balance after the 8th 6 1340.1
year is not yet $1500. 7 1407.1
8 1477.5
9 1551.3
10 1628.9

Y1=1551.32821598

The account will not contain $1500 until
the ninth year. After nine years, the balance
will be $1551.33.

Got It? 4. a. Suppose you invest $500 in a savings account that pays 3.5% annual
interest. When will the account contain at least $650?

b. Reasoning  Use the table in Problem 4 to determine when that account will

contain at least $1650. Explain.

Exponential functions are often discrete. In Problem 4, Account Balance 1600
1500
interest is paid only once a year. So the graph consists of
individual points corresponding to t = 1, 2, 3, and so 1400
on. It is not continuous. Both the table and the graph
1300
show that there is never exactly $1500 in the account
1200
and that the account will not contain more than $1500
1100
until the ninth year.
1000
To model a discrete situation using an exponential 00 2 4 6 8 10
function of the form y = abx, you need to find the Year

growth or decay factor b. If you know y-values for two

consecutive x-values, you can find the rate of change r,
1y2 - y12
and then find b using r = y1 and b = 1 + r.

Lesson 7-1  Exploring Exponential Models 437

Problem 5 Writing an Exponential Function STEM

How can you find the Endangered Species  The table shows the world
value of r? population of the Iberian lynx in 2003 and 2004.
You can use the If this trend continues and the population is
populations for two decreasing exponentially, how many Iberian
consecutive years to lynx will there be in 2014?
find r.
Use the general form of the exponential equation,
How do you find the y = abx = a(1 + r)x.
x-value corresponding
to 2014? Step 1 Define the variables.
The initial x-value
corresponds to 2003, so Let x = the number of years since 2003.
find the difference. Let y = the population of the Iberian lynx.
Step 2 Determine r.

Use the populations for 2003 and 2004.
y2 - y1
r = y1

= 120 - 150 World Population
150 of Iberian Lynx

= -0.2 Year 2003 2004
Population 150 120

Step 3 Use r to determine b.

b = 1 + r = 1 + ( -0.2) = 0.8

Step 4 Write the model.

y = abx Solve for a using the initial
150 = a (0.8)0 values x ϭ 0 and y ϭ 150.
150 = a

The model is y = 150(0.8)x.

Step 5 Use the model to find the population in 2014.
For the year 2014, x = 2014 - 2003 = 11.

y = 150(0.8)x

= 150(0.8)11

≈ 13

If the 2003–2004 trend continues, there will be approximately 13 Iberian lynx in the wild
in 2014.

Got It? 5. a. For the model in Problem 5, what will be the world population of Iberian
lynx in 2020?

b. Reasoning  If you graphed the model in Problem 5, would it ever cross

the x-axis? Explain.

438 Chapter 7  Exponential and Logarithmic Functions

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

Without graphing, determine whether the function PRACTICES
represents exponential growth or exponential decay.
Then find the y-intercept. 7. Vocabulary  E​ xplain how you can tell if y = abx

represents exponential growth or exponential decay.

1. y = 10(0.45)x 2. y = 0.75(4)x 8. Reasoning  ​Identify each function as linear,

quadratic, or exponential. Explain your reasoning.

3. y = 3x 4. y = 0.95x a. y = 3(x + 1)2 b. y = 4(3)x

c. y = 2x + 5 d. y = 4(0.2)x + 1

Graph each function. 6. A (t) = 7(0.6)t 9. Error Analysis  A​ classmate says that the growth
5. A (t) = 3(1.04)t factor of the exponential function y = 15(0.3)x is 0.3.
What is the student’s mistake?

Practice and Problem-Solving Exercises MATHEMATICAL

A Practice Graph each function. PRACTICES See Problem 1.
13. y = 9(3)x
10. y = 6x 11. y = 3(10)x 12. y = 1000(2)x 17. y = 22x
15. s (t) = 1.5t 16. y = 8(5)x
14. f (x) = 2(3)x See Problem 2.

Without graphing, determine whether the function represents exponential

growth or exponential decay. Then find the y-intercept.
( )20. y = 12 ( )21. y
18. y = 129(1.63)x 19. f (x) = 2(0.65)x 17 x = 0.8 1 x
10 8

( ) 22. f(x) = 4 56 x 23. y = 0.45(3)x ( )24. y = 1 4 x 25. f (x) = 2-x
100 3

26. Interest  ​Suppose you deposit $2000 in a savings account that pays See Problems 3 and 4.

interest at an annual rate of 4%. If no money is added or withdrawn

from the account, answer the following questions.
a. How much will be in the account after 3 years?
b. How much will be in the account after 18 years?
c. How many years will it take for the account to contain $2500?
d. How many years will it take for the account to contain $3000?

Write an exponential function to model each situation. Find each amount after See Problem 5.
the specified time.

27. A population of 120,000 grows 1.2, per year for 15 years.

28. A population of 1,860,000 decreases 1.5, each year for 12 years.

29. a. Sports ​Before a basketball game, a referee noticed that the ball seemed
underinflated. She dropped it from 6 feet and measured the first bounce
as 36 inches and the second bounce as 18 inches. Write an exponential
function to model the height of the ball.

b. How high was the ball on its fifth bounce?

Lesson 7-1  Exploring Exponential Models 439

B Apply 30. Think About a Plan  ​Your friend invested $1000 in an account that pays 6% annual
interest. How much interest will your friend have after her college graduation in
4 years?
• Is an exponential model reasonable for this situation?
• What equation should you use to model this situation?
• Is the solution of the equation the final answer to the problem?

STEM 31. Oceanography  ​The function y = 20(0.975)x models the intensity of sunlight
beneath the surface of the ocean. The output y represents the percent of surface
sunlight intensity that reaches a depth of x feet. The model is accurate from about
20 feet to about 600 feet beneath the surface.

a. Find the percent of sunlight 50 feet beneath the surface of the ocean.
b. Find the percent of sunlight at a depth of 370 feet.

32. Population  The population of a certain animal species decreases at a rate of 3.5%
per year. You have counted 80 of the animals in the habitat you are studying.

a. Write a function that models the change in the animal population.
b. Graphing Calculator ​Graph the function. Estimate the number of years until the

population first drops below 15 animals.

33. Sports  While you are waiting for your tennis partner to show up, you drop your
tennis ball from 5 feet. Its rebound was approximately 35 inches on the first bounce
and 21.5 inches on the second. What exponential function would be a good model
for the bouncing ball?

For each annual rate of change, find the corresponding growth or decay factor.

34. +70, 35. +500, 36. -75, 37. - 55,

38. +12.5, 39. -0.1, 40. +0.1, 41. + 100,

C Challenge 42. Manufacturing  The value of an industrial machine has a decay factor of 0.75 per year.
After six years, the machine is worth $7500. What was the original value of the machine?

STEM 43. Zoology  ​Determine which situation best matches the graph. p
A population of 120 cougars decreases 98.75% yearly.

A population of 120 cougars increases 1.25% yearly. 75

A population of 115 cougars decreases 1.25% yearly. 25

A population of 115 cougars decreases 50% yearly. O 10 20 t
30

44. Open-Ended  W​ rite a problem that could be modeled with y = 20(1.1)x.

45. Reasoning  Which function does the graph represent? Explain. (Each interval
represents one unit.)

( ) 1 2x
y= 3

( ) 1 x
y=2 3

( ) y= -2 1 x
3

440 Chapter 7  Exponential and Logarithmic Functions

Standardized Test Prep

SAT/ACT 46. Which function represents the value after x years of a new delivery van that costs
$25,000 and depreciates 15% each year?

y = -15(25,000)x y = 25,000(0.85)x
y = 25,000(0.15)x y = 25,000(1.15)x

47. What is f (x) = 3x 1 for x = 1125 ?
3

15 53 1353 5 13 3

48. What is the simplified form of  22 + i ?
- i

- 1 3 + 4i 5 + 4i 3 + 4i
3 5 5

49. Which graph represents the equation y = x2 - x - 2? 2y

y 2y 2y
2

O 2x Ϫ2 O x O 2x Ϫ2 2x
Ϫ2

E xtended 50. You are driving a car when a deer suddenly darts across the road in front of you.
Response
Your brain registers the emergency and sends a signal to your foot to hit the brake.

The car travels a reaction distance D, in feet, during this time, where D is a function

of the speed r, in miles per hour, that the car is traveling when you see the deer,
11r + 5.
given by D(r) = 10 Find the inverse and explain what it represents. Is the

inverse a function?

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES

MP 1, MP 2

Look back at the information on page 433 about the apparent magnitudes of

stars. Suppose Star 1 is separated from Star 2 on the apparent magnitude scale

by x, and is y times as bright as Star 2.

a. Write an exponential function relating the variables x and y.

b. Graph the exponential function from part (a).

c. Using properties of exponents, rewrite the function in part (a) as an exponential
function with a base of 10. Show your work.

d. Which properties of exponents did you use in part (c) to rewrite the function as an
exponential function with a base of 10?

Lesson 7-1  Exploring Exponential Models 441

7-2 Properties of CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Exponential Functions
MF-IAF.FCS..89 1W2r.Fit-eIFa.3fu.8n ctWiornitdeeafinfuendcbtiyonandexfipnreedssbioyn
ainndeifxfperesnstiobnutineqduifivfearlent fbourtmesq.u.iv. aAllesnot fFo-rImF.Cs ..7.e.,
AF-lBsoF.AM.1AbF,SA.9-C1E2D.F.-AIF.2.3, .A7-eS,SME.AAF.1Sb.912.F-BF.1.1b,
MMAPF1S,.M91P2.2A, -MCEPD3.,1M.2,PM4A, MFSP.951,2M.AP-S7SE.1.1b
MP 1, MP 2, MP 3, MP 4, MP 5, MP 7

Objectives To explore the properties of functions of the form y = abx
To graph exponential functions that have base e

Solve a simpler f and g are exponential functions with the same base. y
problem. Use Is the graph of g f
your calculator to • a compression, g
experiment with • a reflection, or
transformations of • a translation x
y = 2x. of the graph of f? Or is it none of the above? Justify O
your reasoning.

MATHEMATICAL You can apply the four types of transformations—stretches, compressions, reflections,

PRACTICES and translations—to exponential functions.

Essential Understanding  The factor a in y = abx can stretch or compress, and
possibly reflect the graph of the parent function y = bx.

# #The graphs of y = 2x (in red) and y = 3 2x (in blue) are shown. Each y-value of

y = 3 2x is 3 times the corresponding y-value of the parent function y = 2x.

Lesson x y ‫ ؍‬2x y ‫ ؍‬3 · 2x 12 y
10
Vocabulary Ϫ2 1 3 8
• natural base 4 4 6
3
exponential Ϫ1 1 2
function 2
• continuously
compounded 01 3
interest

12 6 4

24 12

x
Ϫ2 O 2 4

#y = 3 2x stretches the graph of the parent function y = 2x by the factor 3.

442 Chapter 7  Exponential and Logarithmic Functions

Problem 1 Graphing y = abx
#How does the graph of
y = − 1 3x compare to the graph of the parent function?
3
Which x-values
Step 1 Make a table of values. Step 2 Graph the function.
should you use to

make a table? x y ‫ ؍‬3x y ‫؍‬ ؊ 1 · 3x Each value is Ϫ 1 times the y
Use x = 0 and then Ϫ2 1 3 3 8
choose both positive 9
Ϫ217 corresponding value of the 6
and negative values.

Ϫ1 1 Ϫ 1 parent function. 4
3 9

01 Ϫ 1 2
3

13 Ϫ1 Ox
29 Ϫ3 Ϫ2 2 4 6

Ϫ2

#
The - 1 in yco=m-p13ress3exsreitflbeycttshethfeacgtroarp31h. of the parent function y = 3x across the
x-axis 3
and The domain and asymptote remain
1
unchanged. The y-intercept becomes - 3 and the range becomes y 6 0.

# Got It? 1. How does the graph of y = -0.5 5x compare to the graph of the parent function?

A horizontal shift y = ab(x-h) is the same as the vertical stretch or compression
y = (ab-h)bx. A vertical shift y = abx + k also shifts the horizontal asymptote
from y = 0 to y = k.

Problem 2 Translating the Parent Function y = bx
How is the graph of
y = 2(x−4) different How does the graph of each function compare to the graph of the parent function?
from the graph of
y = 2x? A y = 2(x−4)
The graph of y = 2(x-4)
is a horizontal translation Step 1 Make a table of values of Step 2 Graph y = 2x then translate
of y = 2x to the right the parent function y = 2x. 4 units to the right.
4 units.
x y ‫ ؍‬2x x y ‫ ؍‬2x y
12
Ϫ2 1 6
4 24 4 4 units

Ϫ1 1 38
2
2
01 Ϫ2 O x
246

The (x - 4) in y = 2(x-4) translates the graph of y = 2x to the right 4 units.
The asymptote remains y = 0. The y-intercept becomes 116.

Lesson 7-2  Properties of Exponential Functions 443

Problem 3

B y = 201 1 2x + 10
2

Step 1 M ake a table of values Step 2 Graph y = 201212x,

for y = 201212x. then translate 10 units up.
y
1 x
2 20
Where have you seen x y ‫؍‬ 20 · A B 10

this situation before? Ϫ1 40 10 units
O 2 4 6x
The graph of a function 0 20
( )liiskbeoyth=a2s0tre12tchx
+ 10 1 10
and a

vertical translation of its 25

parent function. 3 2.5

The “ + 10” in y = 20 1 1 2x + 10 translates the graph of y = 20 1 12 2x up 10 units. It also
2
translates the asymptote, the y-intercept, and the range 10 units up. The asymptote

becomes y = 10, the y-intercept becomes 30, and the range becomes y 7 10. The

domain is unchanged. x

Check Use a graphing calculator to graph y = 20112 2 + 10.



X Y1

0 30
1 20
2 15
3 12.5
4 11.25
5 10.625
6 10.313

X=0

Got It? 2. How does the graph of each function compare to the graph of the
parent function?

# a. y = 4(x+2) b. y = 5 0.25x + 5

Concept Summary  Families of Exponential Functions

Parent function y = bx

Stretch 1 0 a 0 7 12 y = abx

Compression (Shrink) (0 6 0 a 0 6 1) y = b(x-h) + k
y = ab(x-h) + k
Reflection (a 6 0) in x@axis

Translations (horizontal by h; vertical by k)

All transformations combined

444 Chapter 7  Exponential and Logarithmic Functions

Problem 3 Using an Exponential Model STEM Time (min) Temp (°F)
0 203
Why does it make Physics  ​The best temperature to brew coffee is between 5 177
sense that a graph of 195°F and 205°F. Coffee is cool enough to drink at 185°F. The 10 153
this data would have table shows temperature readings from a sample cup of coffee. 15 137
an asymptote? How long does it take for a cup of coffee to be cool enough to 20 121
The temperature of the drink? Use an exponential model. 25 111
hot coffee will get closer 30 104
and closer to room • Set of values Time it takes for Use an exponential
temperature as it cools, • Best serving a cup of coffee to model to find the
but it cannot cool below become cool enough time it takes for
room temperature. temperature to drink coffee to reach 185°F.

Step 1 Step 2
Plot the data to determine if an The graphing calculator exponential model assumes

exponential model is realistic. the asymptote is y = 0. Since room temperature is
about 68°F, subtract 68 from each temperature value.
Calculate the third list by letting L3 = L2 – 68.

The graphing calculator
exponential model assumes
the asymptote is y ϭ 0.

Step 3 Step 4
Use the ExpReg L1, L3 function on Translate y = 134.5(0.956)x vertically by
the transformed data to find an
exponential model. #68 units to model the original data. Use the

model y = 134.5 0.956x + 68 to find how
long it takes the coffee to cool to 185°F.

ExpReg X Y1
y = a1*3b4^.5x169825
a = 2.6 187.65
b = .956011669 2.7 187.11
r2= -.9.9998910685295973691 2.8 186.58
r= 2.9 186.05
3 185.52
3.1 184.99
3.2 184.46

X=3.1

The coffee takes about 3.1 min to cool to 185°F.

Got It? 3. a. Use the exponential model. How long does it take for the coffee to reach
a temperature of 100 degrees?

b. Reasoning ​In Problem 3, would the model of the exponential data be
useful if you did not translate the data by 68 units? Explain.

Lesson 7-2  Properties of Exponential Functions 445

Up to this point you have worked with rational bases. However, exponential functions

can have irrational bases as well. One important irrational base is the number e.
1 2x
The graph of y = 11 + x has an asymptote at y = e or y ≈ 2.71828.

( )x x yy=‫؍‬1A1+؉1x 1xx B x

1 1 y = y2ϭ 2

1010 y ≈y2Ϸ.5924.594 As x approaches infinity
100 y ≈ 2.70 the graph approaches the
1100000 y ≈y2Ϸ.7127.70 value of e.

1000 y Ϸ 2.717

Natural base exponential functions are exponential functions with base e. These
functions are useful for describing continuous growth or decay. Exponential functions
with base e have the same properties as other exponential functions.

Problem 4 Evaluating ex
How can you use a graphing calculator to evaluate e3?

After you press the Method 1 Method 2 Method 3
ex key, what keys Use the ex key. Use the graph of y = ex. Use a table of values for y = ex.
should you press?
e^(3) Y1=e^(X) X Y1
Press 3 , ) , and 20.08553692
01
enter . 1 2.7183
2 7.3891
Xϭ3 Yϭ20.085537 3 20.086
4 54.598
5 148.41
6 403.43

Y1=20.086

e3 ≈ 20.086
Got It? 4. How can you use a graphing calculator to calculate e8?

In Lesson 7-1 you studied interest that was compounded annually. The formula for
continuously compounded interest uses the number e.

Key Concept  Continuously Compounded Interest

amount in account at time t interest rate (annual)

A(t) ϭ P ∙ ert

Principal time in years

446 Chapter 7  Exponential and Logarithmic Functions

Problem 5 Continuously Compounded Interest

What is the Scholarships  Suppose you won a contest at the start of 5th grade that deposited
unknown?
The amount A in the $3000 in an account that pays 5% annual interest compounded continuously.
account after 4 years.
How much will you have in the account when you enter high school 4 years later?

#Express the answer to the nearest dollar. 3664
A = P ert

= 3000e(0.05)(4) Substitute values for P, r, and t. 00000 0
11111 1
= 3000e0.2 Simplify. 22222 2
33333 3
≈ 3664 Use a calculator. Round to the nearest dollar. 44444 4
55555 5
The amount in the account, to the nearest dollar, is $3664. 66666 6
Write your answer, 3664 in the grid. 77777 7
88888 8
99999 9

Got It? 5. About how much will be in the account after 4 years of high school?

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

For each function, identify the transformation from the 5. Vocabulary  I​ s y = e(x+7) a natural base exponential
#parent function y = bx.
function?
1. y = - 2 3x
2. y = 1 (9)x 6. Reasoning  ​Is investing $2000 in an account that pays
2 5% annual interest compounded continuously the
same as investing $1000 at 4% and $1000 at 6%, each
3. y = 7(x-5) 4. y = 5x + 3 compounded continuously? Explain.

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Graph each function. See Problem 1.

7. y = - 5x ( )8. y = 1 x 9. y = 2(4)x
2

10. y = - 9(3)x 11. y = 3(2)x ( )12. y = 24 1 x
2

13. y = - 4x ( )14. y =- 1 x ( )15. y = 23 x
3 2

Graph each function as a transformation of its parent function. See Problem 2.
18. y = - (0.3)x-2
16. y = 2x + 5 ( )17. y = 5 1 x - 8
3

19. y = - 2(5)x+3 20. y = 3(2)x-1 + 4 21. y = - 2(3)x+1 - 5

Lesson 7-2  Properties of Exponential Functions 447

22. Baking  A​ cake recipe says to bake the cake until the Time (min) Temp (°F) See Problem 3.
center is 180°F, then let the cake cool to 120°F. 0 180
The table shows temperature readings for the cake. 5 126 See Problem 4.
10 94 27. ee
a. Given a room temperature of 70°F, what is an 15 80
exponential model for this data set? 20 73

b. How long does it take the cake to cool to the

desired temperature?

Graphing Calculator  Use the graph of y = ex to
evaluate each expression to four decimal places.

23. e6 24. e-2 25. e0 5

26. e2

Find the amount in a continuously compounded account for the given See Problem 5.
conditions.

28. principal: $2000 29. principal: $400 30. principal: $950
annual interest rate: 5.1%
time: 3 years annual interest rate: 7.6% annual interest rate: 6.5%

time: 1.5 years time: 10 years

B Apply 31. Think About a Plan  ​A student wants to save $8000 for college in five years. How

much should be put into an account that pays 5.2% annual interest compounded

continuously?
• What formula should you use?
• What information do you know?
• What do you need to find?

32. Investment  H​ ow long would it take to double your principal in an account that

pays 6.5% annual interest compounded continuously?

( ) 33. Error Analysis  A​ student says that the graph of f (x) = 1 x+2 + 1 is a shift of
3
the parent function 2 units up and 1 unit to the left. Describe and correct the

student’s error.

34. Assume that a is positive and b Ú 1. Describe the effects of c 7 0, c = 0,
and c 6 0 on the graph of the function y = abcx.

35. Graphing Calculator  U​ sing a graphing calculator, graph each of the functions

below on the same coordinate grid. What do you notice? Explain why the definition

of exponential functions has the constraint that b ≠ 1.

( ) ( ) y = 12 x y = 8 x ( ) y = 9 x ( ) y = 99 x
10 10 100

STEM 36. Botany  ​The half-life of a radioactive substance is the time it takes for half of the

material to decay. Phosphorus-32 is used to study a plant’s use of fertilizer. It has a

half-life of 14.3 days. Write the exponential decay function for a 50-mg sample. Find

the amount of phosphorus-32 remaining after 84 days.

STEM 37. Archaeology  A​ rchaeologists use carbon-14, which has a half-life of 5730 years,
to determine the age of artifacts in carbon dating. Write the exponential decay
function for a 24-mg sample. How much carbon-14 remains after 30 millennia?
(Hint: 1 millennium = 1000 years)

448 Chapter 7  Exponential and Logarithmic Functions

The parent function for each graph below is of the form y = abx. Write the
parent function. Then write a function for the translation indicated.

38. y 39. y
Ϫ4 Ϫ2 O
2 x
Ϫ2
4
2 Ϫ4

Ϫ2 O x Ϫ6
24

translation: left 4 units, up 3 units translation: right 8 units, up 2 units

40. Two financial institutions offer different deals to new customers. The first bank
offers an interest rate of 3% for the first year and 2% for the next two years. The
second bank offers an interest rate of 2.49% for three years. You decide to invest the
same amount of principal in each bank. To answer the following, assume you make
no withdrawals or deposits during the three-year period.

a. Write a function that represents the total amount of money in the account in the
first bank after three years.

b. Write a function that represents the total amount of money in the account in the
second bank after three years.

c. Write a function that represents the total amount of money in both accounts at the
end of three years.

STEM 41. Physics  ​At a constant temperature, the atmospheric pressure p in pascals is given 180 ft 210 ft
by the formula p = 101.3e-0.001h, where h is the altitude in meters. What is p at an
altitude of 500 m? Trench is
4 feet wide.
C Challenge 42. Landscaping  ​A homeowner is planting hedges and begins to dig a 3-ft-deep
trench around the perimeter of his property. After the first weekend, the
homeowner recruits a friend to help. After every succeeding weekend, each
digger recruits another friend. One person can dig 405 ft3 of dirt per weekend.
The figure at the right shows the dimensions of the property and the width of
the trench.
a. Geometry ​Determine the volume of dirt that must be removed for the trench.
b. Write an exponential function to model the volume of dirt remaining to be

shoveled after x weekends. Then, use the model to determine how many
weekends it will take to complete the trench.

STEM 43. Psychology  ​Psychologists use an exponential model of the learning process,
f (t) = c(1 - e-kt), where c is the total number of tasks to be learned, k is the
rate of learning, t is time, and f (t) is the number of tasks learned.

a. Suppose you move to a new school, and you want to learn the names of

25 classmates in your homeroom. If your learning rate for new tasks is 20% per

day, how many complete names will you know after 2 days? After 8 days?
b. Graphing Calculator  Graph the function on your graphing calculator. How

many days will it take to learn everyone’s name? Explain.
c. Open-Ended ​Does this function seem to describe your own learning rate?

If not, how could you adapt it to reflect your learning rate?

Lesson 7-2  Properties of Exponential Functions 449

Standardized Test Prep

SAT/ACT 44. A savings account earns 4.62% annual interest, compounded continuously. After
approximately how many years will a principal of $500 double?

2 years 10 years 15 years 44 years

45. What is the inverse of the function f (x) = 1x - 4? f -1(x) = 1x + 4
f -1(x) = x2 - 4, x Ú 0

f -1(x) = x2 + 4, x Ú 0 f -1(x) = 1x - 4
x-4

In Exercises 46 and 47, let f (x) = x2 − 4 and g (x) = x 1 4.
+

46. What is (g ∘ f )(x)? x2 - 1 + 16 - 4 x2 - 4 x - 4
x12 8x x + 4

47. What is ( f ∘ f )(3)? 5 21 77
1

48. What is the equation of the line shown at the right? y
S hort 2
y = - 4 x + 2 -4x + 5y = 7 (2, 3)
Response 5 O
(؊3, ؊1) x
y = 5 x - 2 4x - 5y = 15 24
4

49. How much should you invest in an account that pays 6% annual interest
compounded continuously if you want exactly $8000 after four years?
Show your work.

Mixed Review

Without graphing, determine whether the function represents exponential See Lesson 7-1.
growth or exponential decay. Then find the y-intercept.

50. y = 23(3.03)x 51. f (x) = 3(5)x

( ) 52. y = 2 34 x ( )53. y = 5 8 x
3

Simplify. 55. 23 4 - 2 23 4 See Lesson 6-3.
58. 513 - 2 112 56. 175 + 1125
54. 5 15 + 15 59. 3163 + 128
57. 14 32 + 14 128

Get Ready!  To prepare for Lesson 7-3, do Exercises 60–62.

Find the inverse of each function. Is the inverse a function? See Lesson 6-7.
62. f (x) = 5x3 + 1
60. f (x) = 4x - 1 61. f (x) = x7

450 Chapter 7  Exponential and Logarithmic Functions

7-3 Logarithmic Functions MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
as Inverses
MF-BAFF.SB.941a2  .SFo-lBvFe.2an.4eaq uSaotlivoenaonf ethqeuafotiromn fo(fxt)h=e cfofromr
af(xsi)m=plceffournactsioimnpflethfautnchtaiosnanf tinhvaetrhseasaannd iwnvrietrese. .a.ntdhe
iwnrvieters.e. .AtlhseoinAv-eSrSseE..Al.1sob,MF-AIFF.SC..971e2, .FA-I-FS.SCE.8.1, .F1-bIF,.C.9
MMPAF1S,.M91P2.2F,-MIF.P3.37,eM, MPA4F, SM.9P152.F-IF.3.8,
MAFS.912.F-IF.3.9

MP 1, MP 2, MP 3, MP 4, MP 5

Objectives To write and evaluate logarithmic expressions
To graph logarithmic functions

The chart shows the different ways you can
write 4 and 16 in the form ab, in which
a and b are positive integers and a ≠ 1.

What is the smallest number you can write
in this ab form in four different ways?

In five different ways? In seven different

ways? Explain how you found your answers.

Lesson Many even numbers can be written as power functions with base 2. In this lesson
you will find ways to express all numbers as powers of a common base.
Vocabulary
• logarithm Essential Understanding  The exponential function y = bx is one-to-one, so
• logarithmic its inverse x = by is a function. To express “y as a function of x” for the inverse, write
y = logb x.
function
• common Key Concept  Logarithm

logarithm A logarithm base b of a positive number x satisfies the following definition.
• logarithmic scale For b 7 0, b ≠ 1, logb x = y if and only if by = x.

You can read logb x as “log base b of x.” In other words, the logarithm y is the exponent
to which b must be raised to get x.

The exponent y in the expression by is the logarithm in the equation logb x = y. The
base b in by and the base b in logb x are the same. In both, b ≠ 1 and b 7 0.
Since b ≠ 1 and b 7 0, it follows that by 7 0. Since by = x then x 7 0, so logb x is
defined only for x 7 0.
Because y = bx and y = logb x are inverse functions, their compositions map a
number a to itself. In other words, blogba = a for a 7 0 and logb ba = a for all a.

Lesson 7-3  Logarithmic Functions as Inverses 451

You can use the definition of a logarithm to write exponential equations in
logarithmic form.

T o what power do Problem 1 Writing Exponential Equations in Logarithmic Form
you raise 10 to get
100? What is the logarithmic form of each equation?
10 raised to the 2nd
power equals 100. A 100 = 102

Use the definition of logarithm.
If x = by then logb x = y
If 100 = 102 then log10100 = 2

B 81 = 34

Use the definition of logarithm.
If x = by then logbx = y
If 81 = 34 then log381 = 4

Got It? 1. What is the logarithmic form of each equation? c. 1 = 30

( ) a. 36 = 62 b. 287 = 23 3

You can use the exponential form to help you evaluate logarithms.

Problem 2 Evaluating a Logarithm

Multiple Choice  W​ hat is the value of log8 32? 3 5
53 53

How can you use log8 32 = x Write a logarithmic equation.
32 = 8x Use the definition of a logarithm to write an exponential equation.
the definition of

logarithm to help 25 = (23)x Write each side using base 2.

you find the value of 25 = 23x

log832? Power Property of Exponents
If l=ogbbyx, = y then
x so to what 5 = 3x Since the bases are the same, the exponents must be equal.
Solve for x.
power must you raise 8 5 = x
3
to get 32?
5
Since = 32, then log8 32 = 53.
83

The correct answer is B.

Got It? 2. What is the value of each logarithm? c. log64 1
a. log5 125 b. log4 32 32

452 Chapter 7  Exponential and Logarithmic Functions

A common logarithm is a logarithm with base 10. You can write a common logarithm
log10x simply as log x, without showing the 10.

Many measurements of physical phenomena have such a wide range of values that
the reported measurements are logarithms (exponents) of the values, not the values
themselves. When you use the logarithm of a quantity instead of the quantity, you are
using a logarithmic scale. The Richter scale is a logarithmic scale. It gives logarithmic
measurements of earthquake magnitude.

The Richter Scale
magnitude: ؉ 1

0123456789

E E∙30 E∙302 E∙303 E∙304 E∙305 E∙306 E∙307 E∙308 E∙309

energy released: ؋ 30

Problem 3 Using a Logarithmic Scale

In December 2004, an earthquake with magnitude THAILAND
Magnitude 8.7
9.3 on the Richter scale hit off the northwest coast
MALAYSIA
of Sumatra. The diagram shows the magnitude

of an earthquake that hit Sumatra in March 2005.
iTnhteenfosritmy ulelavellosgoIIf12e=arMth1qu−akMe2s
compares the
where I is the

intensity level determined by a seismograph, Epicenter,
and M is the magnitude on a Richter scale. How March 2005
many times more intense was the December
earthquake than the March earthquake? Sumatra
INDONESIA
log I1 = M1 - M2 Use the formula.
I2

log I1 = 9.3 - 8.7 Substitute M1 = 9.3 and M2 = 8.7.
I2
What is the base of
this logarithm? log I1 = 0.6 Simplify.
This is the common I2
logarithm. It has base 10.
I1 = 100.6 Apply the definition of common logarithm.
I2 Use a calculator.

≈ 4

The December earthquake was about 4 times as strong as the one in March.

Got It? 3. In 1995, an earthquake in Mexico registered 8.0 on the Richter scale. In
2001, an earthquake of magnitude 6.8 shook Washington state. How many
times more intense was the 1995 earthquake than the 2001 earthquake?

Lesson 7-3  Logarithmic Functions as Inverses 453

A logarithmic function is the inverse of an exponential 10 y (1, 10)
function. The graph shows y = 10x and its inverse 8 exponential function
y = log x. Note that (0, 1) and (1, 10) are on the graph of 6 y ‫ ؍‬10x
y = 10x, and that (1, 0) and (10, 1) are on the graph of
y = log x. 4 logarithmic function
y ‫ ؍‬log x
Recall that the graphs of inverse functions are reflections of 2 (10, 1)
each other across the line y = x. You can graph y = logb x (0, 1) x
as the inverse of y = bx.

O 2 4 6 8 10
(1, 0)

Problem 4 Graphing a Logarithmic Function

What is the graph of y = log3x? Describe the domain and range and identify the
y-intercept and the asymptote.

y = log3 x is the inverse of y = 3x.


Step 1 Graph y ϭ 3x. 24 (3, 27) Step 2 Reflecting across
18 y ‫ ؍‬3x the line y ϭ x produces
y‫؍‬x the inverse of y ϭ 3x.

How are the domain 12 (2, 9) y ‫؍‬ log3((2x7), Step 3 Choose a few points on
and range of y = 3x (0, 1) (9, 2) y ϭ 3x and reverse their
and y = log3x 3) coordinates. Plot these
related? new points and graph
Since they are inverse (1, 06) 12 18 24 y ϭ log3 x.
functions, the domain

and range of y = log3x
ra arengthee asnadmdeoamsatihneof The domain is x 7 0. The range is all real numbers. There is no y-intercept.
y = 3x. The vertical asymptote is x = 0.

Got It? 4. a. What is the graph of y = log4 x? Describe the domain, range, y-intercept
and asymptotes.

b. Reasoning ​Suppose you use the following table to help you graph
y = log2x. (Recall that if y = log2x, then 2y = x.) Copy and complete
the table. Explain your answers.

x 2y ‫ ؍‬x y
■
Ϫ1 2y ϭ Ϫ1 ■
0 2y ϭ 0 ■
1 2y ϭ 1 ■
2 2y ϭ 2

454 Chapter 7  Exponential and Logarithmic Functions

The function y = logb x is the parent for a function family. You can graph
y = logb (x - h) + k by translating the graph of the parent function, y = logb x,
horizontally by h units and vertically by k units. The a in y = a logb x indicates a
stretch, a compression, and possibly a reflection.

Concept Summary  Families of Logarithmic Functions

Parent functions: y = logb x, b 7 0, b ≠ 1
y = a logb x
Stretch ( 0 a 0 7 1) y = logb (x - h) + k
Compression (Shrink) (0 6 0 a 0 6 1) s y = a logb (x - h) + k

Reflection (a 6 0) in x-axis
Translations (horizontal by h; vertical by k)
All transformations together

How is the function Problem 5 Translating y = logb x
y = log4(x − 3) + 4 How does the graph of y = log4 (x − 3) + 4 compare to the graph of the
similar to other parent function?
functions you have
seen? Step 1 Step 2

Recall that the graph of Make a table of values for the parent Graph the parent function. Shift the
y = f(x - h) + k is a
vertical and horizontal function. Use the definition of logarithm. graph to the right 3 units and up 4 units

translation of the parent x log4x ‫ ؍‬y 4y ‫ ؍‬x y to graph y = log4 (x - 3) + 4.
function, y = f(x). Ϫ2
4Ϫ2 ϭ 1 Ϫ1 y y ‫ ؍‬log4(x؊3) ؉ 4
1 16 8
16 0
1 4Ϫ1 ϭ 1 4 y ‫ ؍‬log4(x)
4 4 x

1 40 ϭ 1

4 41 ϭ 4 1 O 4 8 12 16 20
16 42 ϭ 16 2 Ϫ4

Because y = log4 (x - 3) + 4 translates the graph of the parent function 3 units to the
right, the asymptote changes from x = 0 to x = 3. The domain changes from x 7 0 to
x 7 3. The range remains all real numbers.

Got It? 5. How does the graph of each function compare to the graph of the
parent function?

a. y = log2(x - 3) + 4 b. y = 5 log2x

Lesson 7-3  Logarithmic Functions as Inverses 455

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

Write each equation in logarithmic form. 9. Vocabulary  ​Determine whether each logarithm is a
common logarithm.
1. 25 = 52 2. 64 = 43
a. log2 4 b. log 64 c. log10 100 d. log5 5
3. 243 = 35 4. 16 = 24
10. Reasoning  E​ xplain how you could use an inverse
Evaluate each logarithm. function to graph the logarithmic function y = log6 x.

5. log2 8 6. log9 9 11. Compare and Contrast  ​Compare the graph of
7. log7 49 8. log2 14 y = log2 (x + 4) to the graph of y = log2 x. How are
the graphs alike? How are they different?

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Write each equation in logarithmic form. See Problem 1.

12. 49 = 72 13. 103 = 1000 14. 625 = 54 15. 110 = 10-1
16. 82 = 64 ( )17. 4 = ( )18. 19. 10-2 = 0.01
1 -2 1 3 = 1
2 3 27

Evaluate each logarithm. See Problem 2.

20. log2 16 21. log4 2 22. log8 8 23. log4 8

24. log2 8 25. log49 7 26. log5 ( - 25) 27. log3 9
28. log2 25 30. log 10,000 31. log5 125
29. log1 1
2
2

STEM Seismology ​In 1812, an earthquake of magnitude 7.9 shook New Madrid, See Problem 3.

Missouri. Compare the intensity level of that earthquake to the intensity level of

each earthquake below.

32. magnitude 7.7 in San Francisco, California, in 1906

33. magnitude 9.5 in Valdivia, Chile, in 1960

34. magnitude 3.2 in Charlottesville, Virginia, in 2001

35. magnitude 6.9 in Kobe, Japan, in 1995

Graph each function on the same set of axes. See Problem 4.
36. y = log2 x 37. y = 2x
38. y = log 1 x ( )39. y =1 x
2 2

Describe how the graph of each function compares with the graph of the parent See Problem 5.
function, y = logb x.

40. y = log5 x + 1 41. y = log7 (x - 2)

42. y = log3 (x - 5) + 3 43. y = log4 (x + 2) - 1

456 Chapter 7  Exponential and Logarithmic Functions

B Apply 44. Think About a Plan  T​ he pH of a substance equals -log[H+], Approximate [H؉] of Foods
where [H+] is the concentration of hydrogen ions, and it ranges
from 0 to 14. A pH level of 7 is neutral. A level greater than 7 is basic, Food [H؉]
and a level less than 7 is acidic. The table shows the hydrogen ion Apple juice
concentration [H+] for selected foods. Is each food basic or acidic? Buttermilk 3.2 ϫ 10−4
• How can you find the pH value of each food? Cream 2.5 ϫ 10−5
• What rule can you use to determine if the food is basic or acidic? Ketchup 2.5 ϫ 10−7
Shrimp sauce 1.3 ϫ 10−4
STEM 45. Chemistry  F​ ind the concentration of hydrogen ions in seawater, if Strained peas 7.9 ϫ 10−8
the pH level of seawater is 8.5. 1.0 ϫ 10−6

Write each equation in exponential form.

46. log2 128 = 7 47. log 0.0001 = -4 48. log6 6 = 1 49. log4 1 = 0
50. log7 16,807 = 5 53. log 10 = 1
51. log2 1 = -1 52. log3 1 = -2
2 9

Find the greatest integer that is less than the value of the logarithm. Use your
calculator to check your answers.

54. log 5 55. log 0.08 56. log 17.52 57. log (1.3 * 107)

58. Compare the graph at the right to the function y = log5 x. Describe the y
domain and range and identify the y-intercept of y = log5 x.
4
59. Write 5 = log2x+1(a + b) in exponential form. (4, 2)
60. Open-Ended  ​Write a logarithmic function of the form y = logb x.
2 (8, 3)
Find its inverse function. Graph both functions on one set of axes.
0 (2, 1) x

2468

Find the inverse of each function.

61. y = log4 x 62. y = log0.5 x 63. y = log10 x 64. y = log2 2x
65. y = log (x + 1) 66. y = log 10x 67. y = log2 4x 68. y = log (x - 6)

Graph each logarithmic function. 71. y = log4 (2x + 3) 72. y = log3 (x + 5)
69. y = log 2x 70. y = 2 log2 x

Find the domain and the range of each function.

73. y = log5 x 74. y = 3 log x 75. y = log2 (x - 3) 76. y = 2 log (x - 2)

You can write 53 = 125 in logarithmic form using the fact that logbbx = x.

log51532 = log5(125) Apply the log base 5 to each side.

3 = log5125 Use logbbx = x to simplify.

Use this method to write each equation in logarithmic form. Show your work.

77. 34 = 81 78. x4 = y 79. 68 = a + 1

Lesson 7-3  Logarithmic Functions as Inverses 457

C Challenge Find the least integer greater than each number. Do not use a calculator.

80. log3 38 81. log1.5 2.5 82. log17 150 83. log5 1
47

84. Match each function with the graph of its inverse.

a. y = log3 x b. y = log2 4x c. y = log1 x
I. II. III. 2

Standardized Test Prep

SAT/ACT 85. Which is the logarithmic form of the exponential equation 23 = 8?

log8 2 = 3 log8 3 = 2 log3 8 = 2 log2 8 = 3

86. Dan will begin advertising his video production business online using a pay-per-click
method, which charges $30 as an initial fee, plus a fixed amount each time the ad
is clicked. Dan estimates that with the cost of 8 cents per click, his ad will be clicked
about 150 times per day. Which expression represents Dan’s total estimated cost of
S hort advertising, in dollars, after x days?

Response (30 + 0.08x)150 360x 30 + 1200x 30 + 12x

87. Which translation takes y = 0 x 0 to y = 0 x + 3 0 - 1? 3 units left, 1 unit down

3 units right, 1 unit down

3 units right, 1 unit up 3 units left, 1 unit up

88. What is the expression 23 ( 1a)7 written as a variable raised to a single rational
exponent?

Mixed Review

Graph each function. 90. y = - 10(4)x+2 See Lesson 7-2.
89. y = 5x - 100 91. y = - 27(3)x-1 + 9

Factor each expression. 93. 4b2 - 100 See Lesson 4-4.
92. 4x2 - 8x + 3 94. 5x2 + 13x - 6

Get Ready!  To prepare for Lesson 7-4, do Exercises 95–98. See Lesson 1-3.
98. x3 + x2; x = 2
Evaluate each expression for the given value of the variable.
# 95. x2 - x; x = 2 96. x3 x5; x = 2
97. xx180; x = 2

458 Chapter 7  Exponential and Logarithmic Functions

Concept Byte Fitting Curves to Data CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

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MP 5

Example 1

Which type of function models the data best—linear,
logarithmic, or exponential?

Connect the points with a smooth curve. Since the points
do not fall along a line, the function is not linear. The graph
appears to approach a horizontal asymptote, so an
exponential function models the data best.

Example 2 xy

Which type of function models the data best—quadratic, 0 14
logarithmic, or cubic? 1 7.5
24
Step 1 Press stat enter to enter the data in lists. 3 1.8
4 1.8
Step 2 Use the stat plot feature to draw a 5 3.9
scatter plot.

Step 3 If you connect the points with a smooth
curve, the end behavior of the graph is
up and up. The graph is not cubic or logarithmic,
so a quadratic function best models the data.

Exercises 2. Which type of function models xy
the data in the table best—linear,
1. Which type of function models the data shown in the quadratic, logarithmic, cubic, or Ϫ1 0
graphing calculator screen best—linear, quadratic, exponential? 1 1.4
logarithmic, cubic, or exponential? 3 2.09
3. Reasoning ​Could you use a 5 2.53
different model for the data in 7 2.81
Exercises 1 and 2? Explain. 9 3.12



Concept Byte  Fitting Curves to Data 459

Example 3

The table shows the number of bacteria in a culture after the given number of Hour Bacteria
hours. Find a good model for the data. Based on the model, how many bacteria 1 Hour220B5acteria
will be in the culture after an additional ten hours? 2 1 22702205
3 2 23502270
Step 1 Press stat enter to enter the data 4 3 26532350
in lists. Use the stat plot feature to 5 4 30522653
draw a scatter plot. 6 5 34173052
7 6 38903417
Step 2 Notice from the scatter plot that the 8 7 45223890
9 8 51074522
data appear exponential. Find the 10 9 57245107
10 5724
equations for the best-fitting
exponential function. Press stat 0
to use the ExpReg feature.

y = 1779.404(1.121)x

Step 3 Graph the function. Press y= clear

vars 5 enter to enter the

ExpReg results. Press graph to display the

function and the scatter plot together.

Press zoom 9 to automatically

adjust the window.

Step 4 In 10 more hours, there will be approximately
y = 1779.404(1.121)20 ≈ 17,474 bacteria in the culture.

Exercises

Use a graphing calculator to find the exponential or quadratic function that best fits
each set of data. Graph each function.

4. x y 5. x y 6. x y 7. x y

Ϫ1 4.9 Ϫ3 0.1 1 3.5 Ϫ1 0.04
0 3.8 Ϫ1 0.4 2 2.11 0 0.1
1 5.0 3 1.30 1 0.5
2 8.1 1 1.6 4 0.73 2 2.5
3 13.3 3 6.4 5 0.28 3 12.5
4 70.2 5 25.6 6 0.08 4 62.5
7 102.4

8. Writing  In Exercise 6 the function appears to level off. Explain why.

9. A savings account begins with $14.00. After 1 year, the account has a balance
of $16.24. After 2 years, the account has a balance of $18.84. Assuming no
additional deposits or withdrawals are made, find the equation for the best-fitting
exponential function to represent the balance of the account after x years. How
much money will be in the account after 20 years?

460 Concept Byte  Fitting Curves to Data

7 Mid-Chapter Quiz athXM

MathXL® for SchoolOL
R SCHO Go to PowerAlgebra.com
L®

FO

Do you know HOW? Graph each logarithmic function. Find the domain
and range.
Determine whether each function is an example of

exponential growth or decay. Then find the y-intercept. 24. y = log5 (x - 1) 25. y = 4 log x + 5

( ) 1. y = 100(0.25)x 1 x = 7 (18)x
2. y = 0.6 10 3. y 8 26. Crafts  For glass to be shaped, its temperature must
stay above 1200°F. The temperature of a piece of
glass is 2200°F when it comes out of the furnace. The
Graph each function. Then find the domain, range, and table shows temperature readings for the glass. Write
y-intercept. an exponential model for this data set and then find
how long it takes for the piece of glass to cool to
4. y = - 4(2)x 5. y = 14(10)x 6. y = 8(0.25)x 1200°F.

7. Investment  ​Suppose you deposit $600 into a savings Time (min) Temp (°F)
account that pays 3.9% annual interest. How much 0 2200
will you have in the account after 3 years if no money
is added or withdrawn? 5 1700

8. Depreciation  ​The initial value of a car is $25,000. 10 1275
After one year, the value of the car is $21,250. Write
an exponential function to model the expected value 15 1000
of the car. Estimate the value of the car after 5 years.
20 850
Graph each function as a transformation of its parent
function. Write the parent function. 25 650

9. y = 3x - 2 10. y = 21(5)x-1 + 4 Do you UNDERSTAND?
11. y = - (0.5)x+3 ( )12. y = x
-6 3 - 10 27. Error Analysis  A​ student claims the y-intercept of
4 the graph of the function y = ab x is the point (0, b).
What is the student’s mistake? What is the actual
evaluate each expression to four decimal places. y-intercept?

13. e5 3 15. e-4 28. Writing  ​Without graphing, how can you tell whether
an exponential function represents exponential
14. e 2 growth or exponential decay?

Find the amount in a continuously compounded 29. Compare and Contrast  ​Compare the graph
account for the given conditions. of y = log3 (x + 1) to the graph of its inverse
y = 3x - 1. How are the graphs alike? How are they
16. principal: $500; annual interest rate: 4.9%; different?
time: 2.5 years
30. Vocabulary  ​Explain how the continuously
17. principal: $6000; annual interest rate: 6.8%; compounded interest formula differs from the
time: 10 years annually compounded interest formula.

Write each equation in logarithmic form.

18. 104 = 10,000 19. 1 = 4-1 ( )20. 8 = 1 -3
4 2

Evaluate each logarithm.

21. log8 64 22. log4 (256) 23. log1 625
5

Lesson 7  Mid-Chapter Quiz 461

7-4 Properties of Logarithms CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

PMrAepFSa.r9e1s2f.oF-rBFF-.L2E.a.AU.4se  Fthoer ecxhpaonngenotifabl amseodfoelrsm, euxlap.ress
aMsPa 1lo,gMarPith2m, MthPe s3olution to abct = d where a, c, and
d are numbers and the base b is 2, 10, or e; evaluate the
logarithm using technology.

MP 1, MP 2, MP 3

Objective To use the properties of logarithms

Write a positive number on a piece of log (Ans) Press enter
paper. Key this number into your calculator 2 * Ans after each step.
and press enter . Then perform the steps Ans + 4
shown here. Press enter after each line. 10^Ans
Do you recognize the number that results? √(Ans)
Explain why this result makes sense. Ans/100

Lesson You can derive the properties of logarithms from the properties of exponents.

Vocabulary Essential Understanding  ​Logarithms and exponents have corresponding
• Change of Base properties.

Formula

Here’s Why It Works  You can use a product property of exponents to derive a
product property of logarithms.

Let x = logb m and y = logb n.

#m = bx and n = by Definition of logarithm

mn = bx by Write mn as a product of powers.

mn = bx+y Product Property of Exponents

logb mn = x + y Definition of logarithm
logb mn = logb m + logb n Substitute for x and y.

Properties  Properties of Logarithms

For any positive numbers m, n, and b where b ≠ 1, the following properties apply.

Product Property logb mn = logb m + logb n
Quotient Property logb mn = logb m - logb n
Power Property logb mn = n logb m

462 Chapter 7  Exponential and Logarithmic Functions

Problem 1 Simplifying Logarithms

What is each expression written as a single logarithm?

A log4 32 − log4 2 32
2
log4 32 - log4 2 = log4 Quotient Property of Logarithms

W hat must you do = log4 16 Divide.
w ith the numbers = log4 42 Write 16 as a power of 4.
that multiply the
logarithms? = 2 Simplify.
Ao fpLpolygathriethPmows.er Property
B 6 log2 x + 5 log2 y

6 log2 x + 5 log2 y = log2 x6 + log2 y5 Power Property of Logarithms
Product Property of Logarithms
= log2 x6y5

Got It? 1. What is each expression written as a single logarithm?
a. log4 5x + log4 3x b. 2 log4 6 - log4 9

You can expand a single logarithm to involve the sum or difference of two or
more logarithms.

Problem 2 Expanding Logarithms

What is each logarithm expanded?

A log 4yx

Can you apply the log 4yx = log 4x - log y Quotient Property of Logarithms
P ower Property of Product Property of Logarithms
LN oog; tahreitfhoumrtshfpiroswt?er = log 4 + log x - log y
a pplies only to x.
B log9 x4
729

log9 x4 = log9 x4 - log9 729 Quotient Property of Logarithms
729

= 4 log9 x - log9 729 Power Property of Logarithms

= 4 log9 x - log9 93 Write 729 as a power of 9.

= 4 log9 x - 3 Simplify.

Got It? 2. What is each logarithm expanded?
a. log3 23570 b. log3 9x5

Lesson 7-4  Properties of Logarithms 463

You have seen logarithms with many bases. The log key on a calculator finds log10
of a number. To evaluate a logarithm with any base, use the Change of Base Formula.

Property  Change of Base Formula

For any positive numbers m, b, and c, with b ≠ 1 and c ≠ 1,

logb m = logc mb .
logc

Here’s Why It Works 

logb m = (logb m)(logc b) Multiply logb m by logc b = 1.
logcb logc b

= logc blogb m Power Property of Logarithms
logcb

= logc m blogb m = m
logc b

Problem 3 Using the Change of Base Formula

W hat common base What is the value of each expression?
has powers that
equal 27 and 81? A log81 27
3; 33 = 27 and
34 = 81. Method 1 Use a common base.

log81 27 = log3 2871 Change of Base Formula
log3

= 3 Simplify.
4

Method 2 Use a calculator.

log81 27 = log 8271 Change of Base Formula log(27)/log(81) .75
log

= 0.75 Use a calculator.

W hat would be a B log5 36
log5 36 = lloogg356
reasonable result? Change of Base Formula log(36)/log(5)
52 = 25 and 53 = 125, ≈ 2.23 Use a calculator to evaluate. 2.226565505

sb oetlwoge5en362 should be
and 3.

Got It? 3. Use the Change of Base Formula. What is the value of
each expression?

a. log8 32 b. log4 18

464 Chapter 7  Exponential and Logarithmic Functions

Problem 4 Using a Logarithmic Scale STEM

Chemistry  T​ he pH of a substance equals −log [H+], where [H+] is the
vcoinnecgeanrtirsa6ti.o3n×of1h0y−d3r.oWgehnatioisntsh.e[Hdi+ffae]rfeonrcheooufstehheoplHd alemvemlsoonfiaamism10o−n1ia1.a[nHd+vv]infoegrar?


Write the equation pH = − log [H+]
for pH.

Write the difference of − log [H +a] − ( − log [H+v])
the pH levels. = − log [H +a] + log [H +v]
= log [H +v] − log [H +a]
Substitute values for
[H+v] and [H+a]. = log (6.3 × 10−3) − log 10−11
Use the Product Property
of Logarithms, and simplify. = log 6.3 + log 10−3 − log 10−11
Use a calculator. = log 6.3 − 3 + 11
? 8.8
Write the answer.
The pH level of ammonia is about
8.8 greater than the pH level
of vinegar.

Got It? 4. Reasoning ​Suppose the hydrogen ion concentration for Substance A is
twice that for Substance B. Which substance has the greater pH level? What
is the greater pH level minus the lesser pH level? Explain.

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Write each expression as a single logarithm.
6. Vocabulary  S​ tate which property or properties
1. log4 2 + log4 8 need to be used to write each expression as a single
2. log6 24 - log6 4 logarithm.

a. log4 5 + log4 5 b. log5 4 - log5 6

Expand each logarithm. 7. Reasoning  I​ f log x = 5, what is the value of 1x?

3. log3 xy 4. log m2n5 5. log2 1x 8. Open-Ended  W​ rite log 150 as a sum or difference of
two logarithms. Simplify if possible.

Lesson 7-4  Properties of Logarithms 465

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Write each expression as a single logarithm. See Problem 1.
11. 5 log 3 + log 4
9. log 7 + log 2 10. log2 9 - log2 3 14. log 5 - k log 2
12. log 8 - 2 log 6 + log 3 13. 4 log m - log n 17. log3 4 + log3 y + log3 8x

15. log6 5 + log6 x 16. log7 x + log7 y - log7 z

Expand each logarithm. See Problem 2.
18. log x3y5
22. log5 rs 19. log7 49xyz 20. logb bx 21. log a2
26. log4 5 1x 23. log3 (2x)2
27. log8 8 23a5 24. log3 7(2x - 3)2 25. log a2b3
28. log5 2x5 c4
29. log 10m4n-2

Use the Change of Base Formula to evaluate each expression. See Problem 3.
33. log5 10
30. log2 9 31. log12 20 32. log7 30 37. log3 33
34. log4 7 35. log3 54 36. log5 62

STEM 38. Science  ​The concentration of hydrogen ions in household dish See Problem 4.
detergent is 10-12. What is the pH level of household dish detergent?

B Apply Use the properties of logarithms to evaluate each expression.

39. log2 4 - log2 16 40. log2 96 - log2 3 41. log3 27 - 2log3 3
42. log6 12 + log6 3 43. log4 48 - 21log4 9
44. 1 log5 15 - log5 175
2

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the loudness of construction work not to exceed 100 dB. Suppose a construction team
is blasting rock for a roadway. One explosion has an intensity of 1.65 * 10-2 W>m2. Is
this explosion in violation of town regulations?
• Which physical value do you need to calculate to answer the question?
• What values should you use for I and I0?

STEM 46. Construction  ​The foreman of a construction team puts up a sound barrier that

reduces the intensity of the noise by 50%. By how many decibels is the noise
I
reduced? Use the formula L = 10 log I0 to measure loudness. (Hint: Find the

difference between the expression for loudness for intensity I and the expression

for loudness for intensity 0.5I.)

47. Error Analysis  ​Explain why the expansion at the right of log4 5st is log4 t = 1 log4 t
incorrect. Then do the expansion correctly. s 2 s

48. Reasoning  ​Can you expand log3 (2x + 1)? Explain. = 1 log4t – log4S
2
# # 49. Writing  E​ xplain why log (5 2) ≠ log 5 log 2.

466 Chapter 7  Exponential and Logarithmic Functions

Determine if each statment is true or false. Justify your answer.

50. log2 4 + log2 8 = 5 51. log3 3 = 1 log3 3
2 2

52. log (x - 2) = lloogg 2x 53. logb x = logb xy
logb y

54. (log x)2 = log x2 55. log4 7 - log4 3 = log4 4

Write each logarithmic expression as a single logarithm.

56. 41 log3 2 + 41 log3 x 57. 21 (logx 4 + logx y) - 3 logx z
58. x log4 m + 1y log4 n - log4 p
( )59. 2 logb x + 3logb y - 5 logb z
3 4

Expand each logarithm. 3

60. log 52yx 61. log s 1t27 62. log a2 1x b 63. log m3
5 n4p-2

64. log 4 54s2r 65. logb 1x 23 y2 66. log4 2x5y7 67. log 2x2 -4
25 z 2 zw4 (x + 3)2

Write each logarithm as the quotient of two common logarithms. Do not

simplify the quotient.

68. log7 2 69. log3 8 70. log5 140 71. log9 3.3 72. log4 3x

STEM Astronomy  ​The apparent brightness of stars is measured

on a logarithmic scale called magnitude, in which lower

numbers mean brighter stars. The relationship between Capella
m = 0.1
the ratio of apparent brightness of two objects and the

difference in their magnitudes is given by the formula
a−ppma1re=nt−br2i.g5hltongebbs12s,.
m2 where m is the magnitude and b is
the

73. How many times brighter is a magnitude 1.0 star than a
magnitude 2.0 star?

74. The star Rigel has a magnitude of 0.12. How many times
brighter is Capella than Rigel?

C Challenge Expand each logarithm. 76. log3 1 + z2]3 77. log7 1r + 9

75. log 5xy122 [(xy3) s2t 1
3

Simplify each expression.

78. log3(x + 1) - log3(3x2 - 3x - 6) + log3(x - 2)

79. log(a2 - 10a + 25) + 1 log(a 1 5)3 - log( 1a - 5)
2 -

Lesson 7-4  Properties of Logarithms 467

Standardized Test Prep

SAT/ACT 80. Which expression is NOT equivalent to 26 16r2?
1 1 (4r)31
116r22 6 4 r 3 23 4r

81. Assume that there are no more turning points beyond those shown. Which graph
CANNOT be the graph of a fourth degree polynomial?

y y

xx

y y
x x

82. A florist is arranging a bouquet of daisies and tulips. He wants twice as many
S hort daisies as tulips in the bouquet. If the bouquet contains 24 flowers, how many
Response daisies are in the bouquet?

8 daisies 12 daisies 16 daisies 24 daisies

83. Use the properties of logarithms to write log 18 in four different ways. Name each
property you use.

Mixed Review

Write each equation in logarithmic form. See Lesson 7-3.

84. 49 = 72 85. 1 = 8-32 86. 5-3 = 1
4 125

Solve. Check for extraneous solutions. See Lesson 6-5.

87. 23 y4 = 16 88. 13 7x - 4 = 0 89. 21w - 1 = 1w + 2

Write a polynomial function with rational coefficients and the given roots. See Lesson 5-5.

90. 13, -5 91. -i, 4i 92. - 17, 1 + 2i

Get Ready!  To prepare for Lesson 7-5, do Exercises 93–95.

Evaluate each logarithm. See Lesson 7-3.

93. log12 144 94. log4 64 95. log64 4

468 Chapter 7  Exponential and Logarithmic Functions

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Logarithmic Equations
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MP 1, MP 3, MP 4, MP 5, MP 7

Objective To solve exponential and logarithmic equations

Make sure you win You are a winner on
the most money. a TV game show.
Which prize would
MATHEMATICAL you choose? Explain.

PRACTICES

Lesson Any equation that contains the form bcx, such as a = bcx where the
exponent includes a variable, is an exponential equation.
Vocabulary
• exponential Essential Understanding  Y​ ou can use logarithms to solve exponential
equations. You can use exponents to solve logarithmic equations.
equation
• logarithmic

equation

Problem 1 Solving an Exponential Equation—Common Base

Multiple Choice  W​ hat is the solution of 163x = 8?

x = 41 x = 73 x = 1 x = 4
What common base is 163x = 8
appropriate?
2 because 16 and 8 are (24)3x = 23 Rewrite the terms with a common base.
both powers of 2.

212x = 23 Power Property of Exponents

12x = 3 If two numbers with the same base are equal, their exponents are equal.
x = 14 Solve and simplify.

The correct answer is A.

Got It? 1. What is the solution of 273x = 81?

Lesson 7-5  Exponential and Logarithmic Equations 469

When bases are not the same, you can solve an exponential equation by

taking the logarithm of each side of the equation. If m and n are positive
and m = n, then log m = log n.

Problem 2 Solving an Exponential Equation—Different Bases

Which property of What is the solution of 153x = 285?
logarithms will help
isolate x? 153x = 285 Take the logarithm of each side.
The rule log ax = x log a Power Property of Logarithms
moves x out of the log 153x = log 285
exponent position. Divide each side by 3 log 15 to isolate x.
3x log 15 = log 285 Use a calculator.
x = 3lolgog28155
x ≈ 0.6958

Check  153x = 285
153(0.6958) ≈ 285.0840331 ≈ 285 ✓

Got It? 2. a. What is the solution of 52x = 130?
b. Reasoning  ​Why can’t you use the same method you used in Problem 1 to

solve Problem 2?

Problem 3 Solving an Exponential Equation With a Graph or Table
What is the solution of 43x = 6000?

How do you choose Method 1 Solve using a graph. WINDOW Intersection
TblStart and ∆Tbl Xmin = –5 Xϭ2.0917911 Yϭ6000
values? Use a graphing calculator. Graph the Xmax = 6
Start with 0 and 1, equations. Xscl = .1 X Y1
respectively. Adjust both Ymin = –500
values as you close in on Y1 = 43x Ymax = 7000 2.05 5042.8
the solution. Y2 = 6000 Yscl = 400 2.06 5256.9
Xres = 1 2.07 5480.2
Adjust the window to find the point 2.08 5712.9
of intersection. The solution is x ≈ 2.09. TABLE SETUP 2.09 5955.5
TblStart = 2.05 2.1 6208.4
Method 2 Solve using a table. 2.11 6472
Tbl = .01
Use the table feature of a graphing Indpnt: Auto Ask Y1=5955.47143094
calculator. Enter Y1 = 43x. Depend: Auto Ask
Use the TABLE SETUP and Tbl
features to locate the x-value that gives
the y-value closest to 6000.
The solution is x ≈ 2.09.

Got It? 3. what is the solution of each exponential equation? check your answer.
a. 74x = 800 b. 5.23x = 400

470 Chapter 7  Exponential and Logarithmic Functions

Problem 4 Modeling With an Exponential Equation STEM

Resource Management  Wood is a sustainable, renewable, natural resource when
you manage forests properly. Your lumber company has 1,200,000 trees. You plan
to harvest 7% of the trees each year. How many years will it take to harvest half of
the trees?

• Number of trees Number of years • Write an exponential equation.
• Rate of decay it takes to harvest • Use logarithms to solve the equation.
600,000 trees

What equation Step 1 Is an exponential model reasonable for this situation?
should you use to Yes, you are harvesting a fixed percentage each year.
model this situation?
Step 2 Define the variables and determine the model.
Since you are planning to
Let n = the number of years it takes to harvest half of the trees.
harvest 7% of the trees Let T(n) = the number of trees remaining after n years.
A reasonable model is T (n) = a(b)n.
each year, you should use Step 3 Use the model to write an exponential equation.
y = abx, where b is the
decay factor.

T(n) = 600,000

a = 1,200,000

r = -7, = -0.07

b = 1 + r = 1 + ( -0.07) = 0.93

So, 1,200,000(0.93)n = 600,000.

Step 4 Solve the equation. Use logarithms.

1,200,000(0.93)n = 600,000

0.93n = 600,000 Isolate the term with n.
1,200,000

log 0.93n = log 0.5 Take the logarithm of each side.

n log 0.93 = log 0.5 Power Property of Logarithms

n = log 0.5 Solve for n.
log 0.93

n ≈ 9.55 Use a calculator.

It will take about 9.55 years to harvest half of the original trees.

Got It? 4. After how many years will you have harvested half of the trees if
you harvest 5% instead of 7% yearly?

A logarithmic equation is an equation that includes one or more logarithms
involving a variable.

Lesson 7-5  Exponential and Logarithmic Equations 471

Problem 5 Solving a Logarithmic Equation

How do you convert What is the solution of log (4x − 3) = 2?
between log form
and exponential Method 1 Solve using exponents.
form?
Use the rule: log a = b log (4x - 3) = 2
if and only if a = 10b.
4x - 3 = 102 Write in exponential form.

4x = 103 Simplify.

x = 103 = 25.75 Solve for x.
4

Method 2 Solve using a graph. Method 3 Solve using a table.
Graph the equations Enter Y1 = LOG (4x − 3).
Y1 = LOG (4x − 3) and Y2 = 2. Use the TABLE SETUP feature to find the
Find the point of intersection.
The solution is x = 25.75. x-value that corresponds to a y-value of

2 in the table.
The solution is x = 25.75.

TABLE SETUP X Y1
TblStart = 25
25.7 1.9991
Tbl = .01 25.71 1.9993
Indpnt: Auto Ask 25.72 1.9995
Depend: Auto Ask 25.73 1.9997
25.74 1.9998
Intersection 25.75 2
Xϭ25.75 Yϭ2 25.76 2.0002

Y1=2

Got It? 5. What is the solution of log (3 - 2x) = -1?

Problem 6 Using Logarithmic Properties to Solve an Equation

What is the domain What is the solution of log (x − 3) + log x = 1?
of the logarithmic
function? log (x - 3) + log x = 1
Logs are defined only for
positive numbers. The log log ((x - 3)x) = 1 Product Property of Logarithms
of a negative number is
undefined. (x - 3)x = 101 Write in exponential form.

x2 - 3x - 10 = 0 Simplify to a quadratic equation in standard form.

(x - 5)(x + 2) = 0 Factor the trinomial.

x = 5  or  x = -2 Solve for x.

Check

log (x - 3) + log (x) = 1 log (x - 3) + log (x) = 1
log ( -2 - 3) + log ( -2) ≟ 1 ✘ log (5 - 3) + log (5) ≟ 1

log 2 + log 5 ≟ 1
If log (x - 3) + log(x) = 1, x = 5.
0.3010 + 0.6990 = 1 ✔

Got It? 6. What is the solution of log 6 - log 3x = -2?

472 Chapter 7  Exponential and Logarithmic Functions

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? 5. Error Analysis  D​ escribe and PRACTICES
Solve each equation. correct the error made in
1. 3x = 9 solving the equation. log2 x = 2 log3 9
2. 2 y+1 = 25 log2 x = log3 92
3. log 4x = 2 6. Reasoning  ​Is it possible for
4. log x - log 2 = 3 an exponential equation to x = 92
have no solutions? If so, give an x = 81
example. If not, explain why.

Practice and Problem-Solving Exercises MATHEMATICAL

A Practice Solve each equation. PRACTICES See Problem 1.

7. 2x = 8 8. 32x = 27 9. 43x = 64 10. 53x = 1
11. 25x+1 = 32 12. 3-2x+2 = 81 13. 23x = 4x+1 125

14. 3x+2 = 272x

Solve each equation. Round to the nearest ten-thousandth. See Problem 2.
Check your answers.
18. 5 - 3x = - 40
15. 2x = 3 16. 4x = 19 17. 8 + 10x = 1008 22. 23x-4 = 5

19. 92y = 66 20. 12y-2 = 20 21. 252x+1 = 144

Graphing Calculator  Solve by graphing. Round to the nearest ten-thousandth. See Problem 3.

23. 47x = 250 24. 53x = 500 25. 6x = 4565 26. 1.5x = 356

Use a table to solve each equation. Round to the nearest hundredth.

27. 2x+3 = 512 28. 3x-1 = 72 29. 62x = 10 30. 52x = 56

31. The equation y = 6.72(1.014)x models the world population y, in billions of See Problem 4.
people, x years after the year 2000. Find the year in which the world population

is about 8 billion.

Solve each equation. Check your answers. 34. log (3x + 1) = 2 See Problem 5.
32. log 2x = -1 33. 2 log x = -1 38. 2 log (x + 1) = 5 35. log x + 4 = 8
36. log 6x - 3 = -4 37. 3 log x = 1.5 39. log (5 - 2x) = 0

Solve each equation. 41. log 2x + log x = 11 See Problem 6.
40. log x - log 3 = 8 44. 3 log x - log 6 + log 2.4 = 9 42. 2 log x + log 4 = 2
43. log 5 - log 2x = 1 45. log (7x + 1) = log (x - 2) + 1

Lesson 7-5  Exponential and Logarithmic Equations 473

B Apply 46. Think About a Plan  ​An earthquake of magnitude 9.1 occurred in 2004 in the

Indian Ocean near Indonesia. It was about 74,900 times as strong as the greatest
earthquake ever to hit Texas. Find the magnitude of the Texas earthquake.
(Remember that an increase of 1.0 on the Richter scale means an earthquake
is 30 times stronger.)
• Can you write an exponential or logarithmic equation?
• How does the solution of your equation help you find the magnitude?

x

47. Consider the equation 23 = 80.
a. Solve the equation by taking the logarithm base 10 of each side.
b. Solve the equation by taking the logarithm base 2 of each side.
c. Writing ​Compare your result in parts (a) and (b). What are the advantages of

each method? Explain.

STEM 48. Seismology  A​ n earthquake of magnitude 7.7 occurred in 2001 in Gujarat, India. It
was about 4900 times as strong as the greatest earthquake ever to hit Pennsylvania.
What is the magnitude of the Pennsylvania earthquake? (Hint: Refer to the Richter
scale on page 453.)

49. As a town gets smaller, the population of its high school decreases by 6% each year. The
senior class has 160 students now. In how many years will it have about 100 students?
Write an equation. Then solve the equation without graphing.

Mental Math ​Solve each equation.

50. 2x = 21 51. 3x = 27 52. log9 3 = x 53. log4 64 = x
54. log8 2 = x 56. log7 343 = x
55. 10x = 1 57. 25x = 1
100 5

58. Demography  T​ he table below lists the states with the highest and with the lowest

population growth rates. Determine in how many years each event can occur. Use
the model P = P0(1 + r)x, where P0 is population from the table, as of July, 2007;
x is the number of years after July, 2007, P is the projected population, and r is the

growth rate.
a. Population of Idaho exceeds 2 million.
b. Population of Michigan decreases by 1 million.
c. Population of Nevada doubles.

Growth Population Growth Population
State rate (%) (in thousands) State rate (%) (in thousands)

1. Nevada 2.93 2,565 46. New York 0.08 19,298

2. Arizona 2.81 6,339 47. Vermont 0.08 621

3. Utah 2.55 2,645 48. Ohio 0.03 11,467

4. Idaho 2.43 1,499 49. Michigan Ϫ0.30 10,072

5. Georgia 2.17 9,545 50. Rhode Island Ϫ0.36 1,058

SOURCE: U.S. Census Bureau

474 Chapter 7  Exponential and Logarithmic Functions

59. Open-Ended  ​Write and solve a logarithmic equation.
60. Reasoning  T​ he graphs of y = 23x and y = 3x+1 intersect at approximately

(1.1201, 10.2692). What is the solution of 23x = 3x+1?
61. Reasoning  I​ f log 120.5x = log 143.6, then 120.5x = ? .

STEM Acoustics ​In Exercises 62–63, the loudness measured in decibels (dB) is
defined by loudness = 10 logII0, where I is the intensity and I0 = 10−12 W>m2.

62. The human threshold for pain is 120 dB. Instant perforation of the eardrum
occurs at 160 dB.

a. Find the intensity of each sound.
b. How many times as intense is the noise that will perforate an eardrum as the

noise that causes pain?

63. The noise level inside a convertible driving along the freeway with its top up is
70 dB. With the top down, the noise level is 95 dB.

a. Find the intensity of the sound with the top up and with the top down.
b. By what percent does leaving the top up reduce the intensity of the sound?

Solve each equation. If necessary, round to the nearest ten-thousandth.

64. 8x = 444 65. 12 log x + log 4 = 2

66. 4 log3 2 - 2 log3 x = 1 67. log x2 = 2

68. 92x = 42 69. log8 (2x - 1) = 1
3

70. log (5x - 4) = 3 71. 124-x = 20

72. 53x = 125 73. log 4 + 2 log x = 6

74. 43x = 77.2 75. log7 3x = 3

Use the properties of exponential and logarithmic functions to solve each
system. Check your answers.

76. b yy -= 24xx-+14 = 0 77. 2x+y = 116 78. b log (2x - y) = 1 2
b 4x-y = log (x + y) = 3 log

C Challenge Solve each equation.

79. log7 (2x - 3)2 = 2 80. log2 (x2 + 2x) = 3 81. 3 log2 4 - 12log2 x = 3
2

( ) STEM 82. Meteorology  In the formula P = P01 h
2
4795, P is the atmospheric pressure in

millimeters of mercury at elevation h meters above sea level. P0 is the atmospheric
pressure at sea level. If P0 equals 760 mm, at what elevation is the pressure 42 mm?

Lesson 7-5  Exponential and Logarithmic Equations 475

STEM 83. Music  ​The pitch, or frequency, of a piano Bassoon
Guitar
#note is related to its position on the n Harp
Cello Violin
keyboard by the function F (n) = 440 212, Bass Viola

where F is the frequency of the sound waves

in cycles per second and n is the number

of piano keys above or below Concert A,

as shown. If n = 0 at Concert A, which

of the instruments shown in the diagram

can sound notes at the given frequency?

a. 590 c. 1440

b. 120 d. 2093 –10 –5 0 5 10

Standardized Test Prep

SAT/ACT 84. The graph at the right shows the translation of the graph of the y

parent function y = 0 x 0 down 2 units and 3 units to the right. x

What is the area of the shaded triangle in square units?

85. What does x equal if log (1 + 3x) = 3?

86. Using the change of base formula, what is the value of x for which
log9 x = log3 5?

87. The polynomial x4 + 3x3 + 16x2 - 19x + 8 is divided by the binomial x - 1.
What is the coefficient of x2 in the quotient?

88. What positive value of b makes x2 + bx + 81 a perfect square trinomial?

PERFORMANCE TASK MATHEMATICAL

Apply What You’ve Learned PRACTICES
MP 5
Look back at the information on page 433 about apparent magnitudes of
stars. In the Apply What You’ve Learned section in Lesson 7-1, you wrote an
exponential function to model the problem on page 433.

a. Use the exponential function you wrote in part (c) on page 441 to write an
exponential equation that you can solve for the difference in apparent magnitude
between Polaris and Sirius.

b. Explain how to use a graphing calculator to solve the equation you wrote in part (a).

c. Solve the equation algebraically, and use a graphing calculator to check your
answer. Round your answer to the nearest hundredth.

476 Chapter 7  Exponential and Logarithmic Functions