Exploring Exponential Models
7-1
Vocabulary
Review
1. Cross out the expressions that are NOT powers.
24
6a 1 7
x3
16x
2. Circle the exponents in the expressions below.
56
4x 2 2 3y24
z2
225a24
x 2y 4
Vocabulary Builder
exponential decay (noun) ek spoh NEN shul dee KAY
Definition: For the function y 5 ab x , if a . 0 and 0 , b , 1, the function
represents exponential decay.
Math Usage: The equation y 5 (0.98)x represents exponential decay.
Main Idea: Something that decreases as time passes is decaying.
Example: A car loses about 15% of its value every year. Its value decays
exponentially.
Use Your Vocabulary
3. Circle the equations that represent exponential decay.
3 x
y 5 2(0.1)x
y 5 2(0.3)x
P 5 P0(1.05)n
N 5 N0(0.5)9
t
7 x
y 5 Q4 R
y 5 Q5 R
y 5 105 ? 1.0001x
y 5 28.0 ? 0.9999n
4. Write numbers to make each equation represent exponential decay.
Answers may vary. Samples are given.
1 n
x
a
b
y 5 3 Q 0.5 R
m 5 0.02
2
Chapter 7
186
Q 5 Q0 a
3
4
b
t
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Related Word: decay (verb)
Problem 1 Graphing an Exponential Function
Got It? What is the graph of y 5 4x ?
5. Complete the following table of values.
x
4x
y
3
43
1
64
2
42
1
16
1
41
1
4
0
40
1
6. Graph the points from your table. Then connect
the points with a smooth curve.
8
y
6
4
2
1
41
4
2
42
16
Summary
4
O
2
x
2
4
Exponential Functions
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
For the function y 5 ab x ,
• if a . 0 and b . 1, the function represents exponential growth.
• if a . 0 and 0 , b , 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 5 0, and the range is y . 0.
7. Write D if the function represents exponential decay. Write G if it represents
exponential growth.
G y 5 3(5x )
G y 5 1 Q7 x R
2
D y 5 35 Q 1 R
3
x
D y 5 (0.25)x
Problem 2 Identifying Exponential Growth and Decay
Got It? Identify y 5 3(4)x as an example of exponential growth or decay. What is
the y-intercept?
8. Compare y 5 3(4)x to the function y 5 ab x . Complete each equation.
a5
3
b5
4
Underline the correct word, expression, or phrase to complete each sentence.
9. The value of b is between 0 and 1 / greater than 1 .
10. So, the function represents exponential growth / decay .
11. The y-intercept is (0, 0) / (1, 0) / (0, 3) .
187
Lesson 7-1
Key Concept
Exponential Growth and Decay
You can model exponential growth or decay with this function.
Rate of growth (r . 0) or decay (r , 0)
Amount after t time periods
A(t) 5 a(1 1 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.
12. Draw a line from the function in Column A to the situation it models in Column B.
Column A
Column B
B(m) 5 300(1 2 0.02)m
You buy 50 mice. Every month, the population increases by 15%.
P(m) 5 50(1.15)m
You have 1 lb of dog food. Your dog eats a quarter of it the next
day, and each day afterward the dog eats a quarter of what’s left.
S(t) 5 8(1.05)t
You get a job paying $8 per hour. Every year, the salary increases
by 5% to keep up with the cost of living.
A(d) 5 1(0.75)d
Your savings account holds $300. For all accounts under $1000,
the bank charges a fee of 2% per month, drawn from the account.
Problem 3
Modeling Exponential Growth
interest. How much will be in the account after five years?
13. An exponential growth model applies to this problem because the amount in the
bank increases / decreases by a fixed percentage of 3.5 % each year.
14. Complete the model.
a 5 the initial amount invested 5
500
r 5 the rate of growth per year written as a decimal 5
A(t) 5 a(1 1 r)t 5
500
t
Q 1 1 0.035 R
15. Evaluate the model for t 5 5.
A(t) 5 a(1 1 r)t
A 5 500(1 1 0.035)5
A 5 500(1.035)5
A 5 $593.84
Chapter 7
188
0.035
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Got It? Suppose you invest $500 in a savings account that pays 3.5% annual
Problem 4 Using Exponential Growth
Got It? Suppose you invest $500 in a savings account that pays 3.5% annual
interest. When will the account contain $650?
16. Complete the model.
x
y
4
573.76
5
593.84
17. Use the TABLE feature on your calculator to
complete the table. Use x for t and y for A(t).
6
614.63
18. The account will contain $650 after 8 years.
7
636.14
8
658.40
a5
500
5 the initial amount invested
r 5 0.035
A(t) 5 a(1 1 r)t 5
500
t
Q 1.035 R
Lesson Check • Do you UNDERSTAND?
Vocabulary Explain how you can tell whether y 5 ab x represents exponential
growth or exponential decay.
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19. y 5 2(3)x represents exponential growth. The value of b in this function is 3 .
x
20. y 5 Q 12R represents exponential decay. The value of b in this function is 12 .
21. Now answer the question. Answers may vary. Sample:
If b S 1, the function represents exponential growth. If 0 R b R 1, the
_______________________________________________________________________
function represents exponential decay.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
exponential function
exponential growth
exponential decay
Rate how well you can model exponential growth and decay.
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189
Lesson 7-1
Properties of
Exponential Functions
7-2
Vocabulary
Review
The dashed figure is the image of the solid figure. Identify each transformation
as a translation, reflection, or dilation.
4
y
2.
4
2
2
2
3.
4
2
x
4
y
4
4
2
2
x
O
2
4
x
4
2
2
2
2
2
4
4
4
translation
dilation
exponential growth (noun) ek spoh NEN shul grohth
Opposite: exponential decay
Definition: For the function y 5 ab x , if a . 0 and b . 1, the function represents
exponential growth.
Example: A savings account earns 3 percent interest annually. The amount
in the account grows exponentially. The equation y 5 0.25(9)x represents
exponential growth.
Use Your Vocabulary
4. Circle the functions that represent exponential growth.
y 5 25(1.02)x
y 5 0.8(2.7)x
y 5 6(9.3)x
5. Cross out the functions that do NOT represent exponential growth.
y 5 2(0.5)x
Chapter 7
y 5 6(x 2 9.3)2
y 5 2(2)x
190
4
reflection
Vocabulary Builder
y 5 2.5(0.75)x
y
y 5 3(7)x
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1.
Problem 1 Graphing y 5 abx
Got It? How does the graph of y 5 20.5 ? 5x compare to the graph of the
parent function?
The graphs of y 5 5x (solid) and y 5 20.5 ? 5x (dashed) are shown at the
right. Underline the correct words or value to complete each sentence.
6. Graphing y 5 20.5 ?
parent function.
5x
y
4
involves a reflection / translation of the
7. The domain of y 5 20.5 ?
the parent function.
2
x
5x
4
is / is not the same as the domain of
2
2
4
2
8. The range of y 5 20.5 ? 5x is / is not the same as the range of the
parent function.
4
9. The y-intercept of y 5 20.5 ? 5x is (0, 1) / (0, 20.5) .
10. Multiplying 5x by 0.5 stretches / compresses the vertical scale.
Problem 2 Translating the Parent Function y 5 bx
Got It? How does the graph of y 5 4(x12) compare to the graph of the parent function?
11. The graph of y 5 4(x12) is a horizontal / vertical translation of y 5 4x two units to
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the left / right .
y
12. The graph at the right shows y 5 4x. Sketch the graph of
y 5 4(x12) on the same set of axes.
6
4
2
4
2
O
x
2
4
Concept Summary Families of Exponential Functions
y 5 bx
Parent Function
Stretch A u au . 1B
Compression (Shrink) A0 , u au , 1B
y 5 ab x
Reflection (a , 0) in x-axis
Translations (horizontal by h; vertical by k)
y 5 b(x2h) 1 k
All transformations combined
y 5 ab(x2h) 1 k
191
Lesson 7-2
13. Each dashed figure is a transformation of each solid figure. Label the transformation
as a vertical stretch, vertical compression, reflection, or translation.
3
3
y
3
y
3
x
1O
1
3
x
x
3
1
3
3
3
1
3
stretch
3
1 O
3
3
3
reflection
or translation
Problem 3
y
y
x
1
3
3
translation
translation
or compression
or compression
Using an Exponential Model
Got It? The initial temperature of a cup of coffee is 203 °F. An exponential model
for the temperature y of the coffee after x minutes is y 5 134.5 ? 0.956x 1 68. How
long does it take for the coffee to reach a temperature of 100 °F?
x
31.6
31.7
31.8
31.9
32.0
y
100.45
100.30
100.16
100.01
99.87
15. It takes about
31.9
minutes to cool to 100 °F.
Key Concept
Continuously Compounded Interest
The formula for continuously compounded interest uses the number e:
amount in account at time t
interest rate (annual)
A(t) P ∙ ert
Principal
Problem 5
time in years
Continuously Compounded Interest
Got It? Suppose you won a contest at the start of 9th grade that deposited $3000 in
an account that pays 5% annual interest compounded continuously. You start college
4 years later, and spend 4 years in college. About how much will be in the account
after 4 years of college?
Chapter 7
192
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14. Enter the function into your graphing calculator. Then complete the table.
8
16. The money is earning interest over
years.
17. Use the justifications at the right to solve the problem.
A 5 P ? er?t
5
3000
?e
Q 0.05 R Q
R
8
Substitute values for P, r, and t.
< $ 4475
Use a calculator. Round to the nearest dollar.
Lesson Check • Do you UNDERSTAND?
Reasoning Is investing $2000 in an account that pays 5% annual interest
compounded continuously the same as investing $1000 at 4% and $1000 at 6%,
each compounded continuously? Explain.
18. Cross out the equation(s) that you could NOT use to solve this problem.
A 5 2000 ? e(0.05)t
A 5 1000 ? e(0.05)2t
A 5 1000 A e(0.05)t 1 e(0.06)t B
A 5 1000 ? e(0.04)t
A 5 1000 ? e(0.06)t
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19. Use a calculator to complete the table. Round to the nearest dollar.
t
2000 ∙ e0.5t
1000 ∙ e0.4t
1000 ∙ e0.6t
1000 ∙ e0.4t 1000 ∙ e0.6t
1
$ 2102
$ 1041
$ 1062
$ 2103
3
$ 2324
$ 1128
$ 1197
$ 2325
5
$ 2568
$ 1221
$ 1350
$ 2571
20. Now answer the question. Explanations may vary. Sample:
No. One account at 5% earns less than two accounts at 4% and 6%.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
natural base exponential function
continuously compounded interest
Rate how well you can graph exponential functions with base e.
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193
Lesson 7-2
Logarithmic Functions
as Inverses
7-3
Vocabulary
Review
1. Circle the base in each power.
24
522
(x 2 2)3
3x 5
3x
Vocabulary Builder
logarithm (noun)
uh rith um
LAWG
Related Words: base, power, exponent
Definition: The logarithm base b of a positive number x is the exponent to which
base b must be raised to get x.
Examples: Since 24 5 16, the logarithm base 2 of 16 is 4.
Math Usage: A logarithm is written as log b x and is read “log base b of x.”
Use Your Vocabulary
Write each logarithm in words. The first one is done for you.
Sample: log 2 16
log base 2 of 16
3. log 4 3
2. log 3 4
log base 3 of 4
4. log x y
log base 4 of 3
Key Concept
log base x of y
Logarithm
A logarithm base b of a positive number x satisfies the following definition.
log b x 5 y if and only if b y 5 x
Chapter 7
194
5. log y x
log base y of x
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Similarly, since 43 5 64, the logarithm base 4 of 64 is 3.
6. Draw a line from each logarithm equation in Column A to its exponential equation
in Column B.
Column A
Column B
log 2 16 5 4
103 5 1000
log 3 9 5 2
by 5 x
log 10 1000 2 3
32 5 9
log b x 5 y
24 5 16
Problem 1 Writing Exponential Equations in Logarithmic Form
Got It? What is the logarithmic form of the equation 36 5 6 2 ?
7. Use the definition of logarithm: If x 5 b y , then log b x 5 y .
2
Since 36 5 6
, log 6 36 5
2 .
8. The logarithmic form of the equation is
log6 36 5 2
.
Problem 2 Evaluating a Logarithm
Got It? What is the value of log 5 125?
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9. Use the justifications at the right to complete the equations.
x 5 log 5 125
125 5 5 x
5
Write the logarithm.
Use the definition of a logarithm to write an exponential equation.
3
5 5x
Write 125 as a power of 5.
3
5x
Since the bases are the same, the exponents must be equal.
log 5 125 5
3
Write the value of log 5 125.
A common logarithm is a logarithm with base 10. You can write a common logarithm
log10 x simply as log x, without showing the 10.
Problem 3 Using a Logarithm Scale
Got It? 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?
I
Use the formula log I1 5 M1 2 M2 to compare the intensity levels of earthquakes,
2
where I is the intensity level and M is the magnitude on the Richter scale.
10. Circle the equation that models the problem.
I
log I1 5 2001 2 1995
2
I
log I1 5 8.0 2 6.8
2
195
I
log I1 5 101.2 2 106
2
Lesson 7-3
11. Simplify the equation you circled in Exercise 10.
I
log I1 5 1.2
2
12. Circle the common logarithm that corresponds to the simplified equation.
I1
6
I2 5 10
I1
1.2
I2 5 10
I1
24.8
I2 5 10
13. Use a calculator. The 1995 earthquake was about 15.85 times more intense than
the 2001 earthquake.
Problem 4
Graphing a Logarithmic Function
Got It? What is the graph of y 5 log 4 x? Describe the domain, range, y-intercept,
and asymptotes.
14. The graph of y 5 4 x is at the right.
y
Circle the graph of its inverse y 5 log 4 x.
y
y
24
O
16
8
8
16
8
16
x
24
8
16
x
24
4
O
x
4
8
12
24
15. Draw a line from each item in Column A to a corresponding item in Column B.
Column A
Column B
Domain of y 5 log 4 x
x50
Range of y 5 log 4 x
x.0
y-intercept of graph of y 5 log 4 x
y.0
Vertical asymptote(s) of y 5 log 4 x
none
all real numbers
Concept Summary Families of Logarithmic Functions
Parent function
y 5 log b x, b . 0, b 2 1
Stretch (u a u . 1)
Compression (shrink) (0 , u a u , 1) ¶
Reflection (a , 0) in x-axis
y 5 a log b x
Translations (horzontal by h; vertical by k)
y 5 log b (x 2 h) 1 k
All transformations together
y 5 a log b (x 2 h) 1 k
Chapter 7
8
196
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O
12
16. Circle each equation that is a compression of the parent function. Underline each
equation that is a reflection of the parent function.
y 5 0.2 log8 x
y 5 23 log 4 x
y 5 log 3(x 2 5)
y 5 15 log10 x
Problem 5 Translating y ⴝ logb x
Got It? How does the graph of y 5 logb (x 2 3) 1 4 compare to
the graph of the parent function?
17. The graph at the right shows the parent function, y 5 log 2 x.
Underline the correct word or value to complete each sentence.
To graph y 5 log 2 (x 2 3) 1 4, translate the graph of the parent
function 2 / 3 / 4 units left / right and 2 / 3 / 4 units up / down .
The domain becomes x . 24 / 3 . The range does / does not change.
8
y
4
O
y log2 x
x
4
8
12
16
4
Lesson Check • Do you UNDERSTAND?
Vocabulary Determine whether each logarithm is a common logarithm.
log2 4
log 64
log10 100
log 5 5
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18. Circle the base of each logarithm above.
Complete.
19. Common logarithms have a base of 10 .
20. Logarithms without a base are assumed to have a base of 10 .
21. Write Y for yes or N for no to indicate whether each logarithm is a common logarithm.
N log 4
2
Y log 64
Y log 100
10
N log 5
5
Math Success
Check off the vocabulary words that you understand.
logarithm
logarithmic function
common logarithm
logarithmic scale
Rate how well you can use and graph logarithms.
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197
Lesson 7-3
Properties of Logarithms
7-4
Vocabulary
Review
1. Circle the logarithms below.
39
log4 64
log (x 2 2 1)
a ? b(x11)
2. Circle the base of each logarithm
log2 32
log7 49
log10 57
log5 125
F
3. If logb x 5 y, then b x 5 y.
T
4. If log x 5 w, then 10w 5 x.
F
5. If 32 5 x, then log2 x 5 3.
Vocabulary Builder
formula (noun)
FAWRM
yoo luh
Other Word Forms: formulate (verb), formulaic (adjective)
Definition: A mathematical formula is an equation that you can use to solve a
particular kind of problem.
Example: You can use the quadratic formula to solve quadratic equations. You can
use the distance formula to find the distance between two points.
Use Your Vocabulary
6. Circle the slope formula.
1
A 5 2 (b1 1 b2)h
Chapter 7
A 5 Pert
P 5 2l 1 2w
198
y2 2 y1
m5x 2x
2
1
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Write T for true or F for false.
Since y 5 logax if and only if x 5 a y , logarithms and exponents have corresponding
properties.
Properties Properties of Logarithms
For any positive numbers m, n, and b where b 2 1, the following properties apply.
Product Property
Quotient Property
Power Property
logb mn 5 logb m 1 logb n
m
logb n 5 logb m 2 logb n
logb mn 5 n logb m
Use the properties of logarithms to complete each equation.
7. log 20 5 log (5 ? 4) 5 log 5 1 log 4
8. log5 (x 9) 5
9 log5x
9. log2 24 2 log2 3 5 log2
24
5 log 2 8
3
10. log3(5x 4) 5 log3 5
1 log3 x 4 5 log3 5
1 4 log3x
Problem 1 Simplif ying Logarithms
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Got It? What is 2 log 4 6 2 log 4 9 written as a single logarithm? If possible, simplify
the single logarithm.
11. Circle the property you can use to rewrite 2 log 4 6.
Product Property
Quotient Property
Power Property
12. Use the property you circled above to rewrite the first term of 2 log 4 6 2 log 4 9.
2 log 4 6 2 log 4 9 5 log 4 36 2 log 4 9
13. Circle the property you can use to combine the last two terms in Exercise 12.
Product Property
Quotient Property
Power Property
14. Use the property you circled above to combine the two terms.
log 4 36 2 log 4 9 5 log 4
36
5 log 4 4
9
15. Use the definition of logarithm to simplify the expression.
log 4 4
5
Problem 2
1
Expanding Logarithms
Got It? What is log3 250
37 expanded? Simplify your answer, if possible.
199
Lesson 7-4
16. Follow the steps to expand the logarithm.
log3 250
37 5 log3
2 log3
250
37
Use the Quotient Property of Logarithms.
5 log3 2 1 log3
125
2 log3
37
Use the Product Property of Logarithms.
5 log3 2 1 log3
53
2 log3
37
Write 125 as a power of 5.
5 log3 2 1
3 log3 5 2 log3
37
Use the Power Property of Logarithms.
Properties Change of Base Formula
log m
For any positive numbers m, b, and c, with b u 1 and c u 1, logb m 5 logc b .
c
Use the Change of Base Formula to complete each equation.
log 100
17. log5 100 5
18. log2 100 5
log 5
log5 100
19. log
log5 2
log2 100
100 5 log 3
3
2
20. Reasoning The base implied in Exercise 17 is 2 / 5 / 10 / 100 .
Using the Change of Base Formula
Problem 3
Got It? What is the value of log8 32?
21. Circle the least common factor of 8 and 32.
4
8
32
22. Complete each equation.
Since 25 5 32, log2 32 5
log2 32
23. log8 32 5
log
2
8
5
Since 23 5 8, log2 8 5 3 .
5 .
5
3
Got It? What is the value of log 4 18?
24. Circle the calculator-ready form of log 4 18.
log 4
log 18
log 9
log 2
25. The value of log 4 18 is approximately
Problem 4
log 9
log 4
2.085
log 18
log4
.
Using a Logarithmic Scale
Got It? Chemistry The pH of a substance equals 2log fH1g, where fH1g is
the concentration of hydrogen ions. Suppose the hydrogen ion concentration for
Substance A is twice that for Substance B. Which substance has a greater pH level?
What is the greater pH level minus the lesser pH level? Explain.
Chapter 7
200
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2
26. If fH 1bg is the concentration of hydrogen ions for Substance B, circle the pH
of Substance B.
2 log fH 1bg
log 12 fH 1bg
log f2H 1bg
2log fH 1bg
27. Circle the expression for the concentration of hydrogen ions for Substance A.
fH 1bg 2 2
fH1bg
2
2 ? fH 1bg
log fH 1bg
28. Circle the expression for the pH of Substance A.
2log f2 ? fH 1bgg
212 log fH 1bg
2 log f2fH 1bgg
2log fH 1bg
29. Use the Product Property of Logarithms to expand the expression you circled above.
2log f2 ? fH1b gg
5 2Q log 2 1 log fH 1bg R
5 2 0.301 2 log fH 1bg
(pH of Substance A) 5 2 0.301 1 (pH of Substance B)
30. Circle the substance with the greater pH level.
Substance A
Substance B
31. What is the difference between the pH levels? Explain how you know.
log 2, or 0.301. Explanations may vary. Sample: The log of twice a
_______________________________________________________________________
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quantity x is log 2 more than log x.
_______________________________________________________________________
Lesson Check • Do you UNDERSTAND?
Reasoning If log x 5 5, what is the value of 1x ?
32. Underline the correct expression to complete each sentence.
If log x 5 5, then x 5 105 > 510 . Since 1x 5 x 0 > x 21 , I know 1x 5 1025 > 5210 .
Math Success
Check off the vocabulary words that you understand.
logarithm
Change of Base Formula
Rate how well you can use the properties of logarithms.
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201
Lesson 7-4
Exponential and
Logarithmic Equations
7-5
Vocabulary
Review
1. Underline the base and circle the exponent in each expression.
37
(x 1 y)2
2x 2
3 1 9(2x17)
2. Multiple Choice Which expression is read “five to the third power”?
533
53
5x 3
35
Vocabulary Builder
isolate (verb)
EYE
soh layt
Other Word Forms: isolation (noun), isolated (adjective)
Math Usage: To isolate a variable in an equation means to get the variable alone on
one side of the equal sign.
Use Your Vocabulary
3. Draw a line from each equation in Column A to the operation(s) needed to isolate
the variable in Column B.
Column A
Column B
3b 5 15
Multiply both sides by 3. Then add 5 to both sides.
4z 1 9 5 81
Divide both sides by 3.
1
3(p 2 5) 5 2
Subtract 9 from each side, then divide both sides by 4.
Problem 1
Solving an Exponential Equation—Common Base
Got It? What is the solution of 273x 5 81?
4. Circle the common base that you can use to rewrite 27 and 81.
2
Chapter 7
3
6
202
9
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Definition: When you isolate something, you get it by itself.
5. Rewrite 27 and 81 using the common base.
3 b
a 3
3x
5 3
4
6. If two numbers with the same base are equal, their exponents are
equal / not equal .
7. Solve for x.
A33 B 3x 5 34
39x 5 34
9x 5 4
x 5 49
Problem 2 Solving an Exponential Equation—Different Bases
Got It? What is the solution of 52x 5 130?
8. The equation has been solved below. Write a justification for each step.
52x 5 130
Write the original equation.
__________________________________________________
log 52x 5 log 130
Take the logarithm of each side.
__________________________________________________
2x log 5 5 log 130
Power Property of Logarithms
__________________________________________________
log 130
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
x 5 2 log 5
Divide each side by 2 log 5 to isolate x.
__________________________________________________
9. Use a calculator to determine the value of x to four decimal places.
x < 1.5122
Problem 3 Solving an Exponential Equation With a Graph or Table
Got It? What is the solution of 74x 5 800?
10. Write the two equations you can graph on your graphing calculator to solve
this equation.
Y1 5
74x
Y2 5
800
11. Use a graphing calculator to find the point of intersection of the two equations.
Write the approximate value of x.
x < 0.8588
203
Lesson 7-5
Problem 4 Modeling With an Exponential Equation
Got It? 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 5% of the trees each year. How many years will it take to harvest half
of the trees?
12. You are harvesting a fixed number / percentage of trees each year, so using a
linear / exponential growth / exponential decay model is reasonable.
13. Use the phrases at the right to label each part of the formula.
amount after n periods
rate of decay
initial amount
number of time periods
amount after n periods
number of time periods
n
T(n) a(1 r)
initial amount
rate of decay
14. Use the information in the problem to write each amount.
a5
20.05
r5
1,200,000
T(n) 5
600,000
15. Use your values for a, r, and T(n) to write an equation.
600,000 5 1,200,000(1 2 0.05)n
600,000 5 1,200,000(1 2 0.05)n
600,000
1,200,000
5 (0.95)n
log 0.5 5 n log 0.95
log 0.5
log 0.95
5n
n N 13.5
Problem 5
Solving a Logarithmic Equation
Got It? What is the solution of log (3 2 2x) 5 21?
17. Circle the equivalent exponential form of log (3 2 2x) 5 21.
3 2 2x 5 (21)
10(322x) 5 1021
2log(3 2 2x) 5 1
18. Solve the exponential equation for x.
3 2 2x 5 1021
3 2 2x 5 0.1
22x 5 22.9
x 5 22.9
22 5 1.45
Chapter 7
204
3 2 2x 5 1021
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16. Solve your equation for n, the approximate number of years it will take to harvest half
the original trees.
Problem 6
Using Logarithmic Properties to Solve an Equation
Got It? What is the solution of log 6 2 log 3x 5 22?
19. You can use the Product / Quotient / Power Property of Logarithms
to write an equivalent equation with a single logarithm.
20. Write the single logarithm equation equivalent to log 6 2 log 3x 5 22.
log 2x 5 22
21. Solve the single logarithm equation for x.
log 2x 5 22
2
x
2
x
5 1022
1
5 100
200 5 x
Lesson Check • Do you UNDERSTAND?
Error Analysis Describe and correct the error made in solving the equation.
22. If logz x 5 logb y, does x 5 y? Explain.
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Answers may vary. Sample: Only when the bases are equal (z 5 b)
________________________________________________________________
log2 x = 2 log3 9
log2 x = log3 92
x = 92
x = 81
will x 5 y.
________________________________________________________________
23. Complete the equation below.
2 log3 9 5 2 log3 3
5
4 log3 3
5
4
24. Correct the error and solve for x.
2
log2 x 5 2 log3 9
log2 x 5 4
x 5 24
x 5 16
Math Success
Check off the vocabulary words that you understand.
exponential equation
logarithmic equation
Rate how well you can solve exponential and logarithmic equations.
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Lesson 7-5
Natural Logarithms
7-6
Vocabulary
Review
Write T for true or F for false.
T
1. The function y 5 logb x, where b . 0 and b 2 1 is called a logarithmic function.
F
2. A logarithmic equation is an equation that contains only one logarithm.
F
3. The logarithm of a power is the difference of the logarithm and the exponent.
Vocabulary Builder
inverse function (noun)
IN
vurs
FUNGK
shun
Definition: To find the inverse function, switch the order of the elements in the
ordered pairs of the function.
Example: function, f (x): {(1, 2), (3, 4)}; inverse function, f 21 (x): {(2, 1), (4, 3)}
Use Your Vocabulary
4. Complete the table of values for the inverse function, f 21 (x), of the function f(x).
f1(x) x 3
f(x) x 3
x
y
Chapter 7
1
0
1
2
2
3
4
6
x
2
3
4
6
y
1
0
1
2
206
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Related Words: function, inverse, input, output
Key Concept
Natural Logarithmic Function
If y 5 e x , then x 5 loge y 5 ln y. The natural logarithmic function
is the inverse of y 5 e x , so you can write it as y 5 ln x.
5. If y 5 e 5 , then ln y 5
1
1 y = ex
5 .
6. If ln b 5 6, then b 5 e
6
2
2 y = ln x
.
Problem 1 Simplifying a Natural Logarithmic Expression
Got It? What is ln 7 1 2 ln 5 written as a single natural logarithm?
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
7. The expression is simplified below. Write a justification for each step.
ln 7 1 2 ln 5
Write the original expression.
ln 7 1 ln 52
Power Property of Logarithms
ln 7 1 ln 25
Simplify the second term.
ln (7 ? 25)
Product Property of Logarithms
ln 175
Multiply.
Problem 2 Solving a Natural Logarithmic Equation
Got It? What are the solutions of ln x 5 2?
8. Complete: If ln x 5 2, then e
2
5 x.
Got It? What are the solutions of ln (3x 1 5)2 5 4? Check your answers.
9. The equation is solved below. Write a justification for each step.
ln (3x 1 5)2 5 4
Write the original equation.
ln (3x 1 5)2 5 e4
Write in exponential form.
ln 2(03x 1 5 5 4e2
Take the square root of each side.
ln (0 1 523x 5 25 4 e2
Subtract 5 from each side.
ln (0 1 523x 5 (25 4 e2) 4 3
Divide each side by 3.
ln (0 1 523x < 0.7964 or x < 24.1267
Use a calculator.
207
Lesson 7-6
10. Check Substitute your values for x in ln (3x 1 5)2 5 4. Use a calculator.
ln Q 3 ?
2
1 5R <
0.7964
4.0001
2
ln Q 3 ? 24.1267 1 5 R <
3.9976
Got It? What are the solutions of ln 2x 1 ln 3 5 2? Check your answers.
11. Circle the property of logarithms that justifies writing ln 2x 1 ln 3 as ln 6x.
Power Property
Product Property
12. Use the simplified equation to solve for x.
Quotient Property
13. Check Substitute your value for x in ln 2x 1 ln 3 5 2.
Use a calculator.
ln 6x 5 2
6x 5 e2
ln 2 ?
1.2315
1 ln 3 <
1.9522
e2
x5 6
x N 1.2315
Solving an Exponential Equation
Problem 3
Got It? What is the solution of e x22 5 12? Check your answer.
14. Use the justifications at the right to solve the equation.
e x22 5 12
Write the original equation.
ln 12
x5
ln 12
x<
Rewrite in logarithmic form.
12
Add 2 to each side.
4.4849
Use a calculator.
15. Check Substitute your value for x in e x22 5 12. Use a calculator.
e 4.4849 22 < 11.9999
Got It? What is the solution of 2e 2x 5 20? Check your answer.
16. Circle the first step in solving the equation. Underline the second step.
Divide each side of the equation by 2.
Use the Power Property.
Write in exponential form.
Write in logarithmic form.
17. Now solve the equation.
2e2x 5 20
e2x 5 10
2x 5 ln 10
x 5 2ln 10 5 22.3026
18. Check Substitute your value for x in 2e2x 5 20. Use a calculator.
2e 2
Chapter 7
22.3
< 19.9484
208
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x225
Problem 4 Using Natural Logarithms
Got It? Space A spacecraft can attain a stable orbit 300 km above Earth if it reaches
a velocity of 7.7 km/s. The formula for a rocket’s maximum velocity v in kilometers per
second is v 5 20.0098t 1 c ln R. The booster rocket fires for t seconds and the velocity
of the exhaust is c km/s. The ratio of the mass of the rocket filled with fuel to its mass
without fuel is R.
Suppose a booster rocket for a spacecraft has a mass ratio of about 15, an exhaust
velocity of 2.1 km/s, and a firing time of 30 s. Can the spacecraft achieve a stable orbit
300 km above Earth?
19. Identify the values for each variable.
R 5 15
c 5 2.1
t 5 30
20. Complete the equation for finding the spacecraft’s maximum velocity. Then use a
calculator to simplify the equation.
v 5 20.0098 ? 30 1 Q 2.1 ln 15 R km/s < 5.3929 km/s
21. The spacecraft will / will not achieve a stable orbit 300 km above Earth. Explain.
Answers may vary. Sample: The velocity of 5.4 km/s is less than the
_______________________________________________________________________
7.7 km/s needed to attain a stable orbit.
_______________________________________________________________________
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Lesson Check • Do you UNDERSTAND?
Reasoning Can ln 5 1 log2 10 be written as a single logarithm? Explain your reasoning.
22. The ln 5 1 log2 10 can be written as ln 5 5 loge 5
1 log2 10.
23. Can you use the Product Property of Logarithms to combine the logarithms?
Explain why or why not.
No; the two logarithms have different bases.
_______________________________________________________________________
24. ln 5 1 log2 10 can / cannot be written as a single logarithm.
Math Success
Check off the vocabulary words that you understand.
function
logarithm
natural logarithmic function
Rate how well you can write and solve equations with natural logarithms.
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Lesson 7-6