Lesson 5.2’ Properties of Exponents and Power Functions
Name
Period
Date
1. Rewrite each expression as a fraction without exponents or as an
integer. Using your calculator, verify that your answer is equivalent to
the original expression.
a.
32
b.
d.
(_5)_3
e.
73
c.
(3)_2
_44
(5)_2
2. Rewrite each expression in the form x or ax’
.
1
a. 4x°
8
9x
b. (8X_6)(_15x14)
d —88x’°
3
—8x
9
(—35x
73
\ —7x!
83
(4Ox
\—8x
)
2
3. Solve.
a. 2c
b.
=
125c
=
(4)X =
25
c.
4. Solve each equation for positive values of x. If answers are not exact,
approximate to two decimal places.
a. 6x’
5
d. 8x
9
28
=
=
80
6
6x
CHAPTER 5
b.
2Oxh/2
3
e. 15x
—
=
8
=
4.5
2
10x
c. 5x”
=
f. 200x’
=
0.06
3
125x
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LESSON 5.2 Properties of Exponents
and Power Functions
c.-
b.
d
e.
125
8
2. a. 36x
15
e. 125x
36
_25
•
d. 11x
7
18
° c. x
2
b. —120x
f.
x18, or
1
_
8
—O.008x’
c. x
—2
—5
b. x
4. a. x
5.62
b. x
0.39
c. x
578703.70
d.x
0.91
e. x
1.5
f. x
0.79
3. a. x
=
=
=
Lesson 5.3
•
Rational Exponents and Roots
Name
Period
flltA
1. Identify each function as a power function, an exponential function,
or neither of these. (The function may be translated, stretched, or
reflected.)
a. f(x)
=
3
0.5x
—
4
b.f(x)
c.f(x)+2
2. Rewrite each expression in the form bx in which x is a rational
exponent.
b(\/’)
C.
3. Solve each equation for positive values of x. If answers are not exact,
approximate to the nearest hundredth.
a.=27
d.
—
23
b.
=
—15
e.
=
0.77
8.5
c, 4 + 18
19.8
f.
=
=
32
12.75
4. Each of the following graphs is a transformation of the power function
y = x3/2. Write the equation for each curve.
x
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CHAPTER 5
29
LESSON 5.3
Rational Exponents and Roots
1. a. Power
b. Exponential
c. Power
2. a.
b. d
3
/
4
C. r/2
c3/2
3. a. x
243
b. x
1.69
C. x
42.88
d. x
=
32
e. x
18.99
f.
58.72
4. a. y
=
x3/2
—
b. y
_x3/2
C.
(x +
y
d. y
=
e. y
1. y
3
2)3/2
4 + (x
—
3)3/2
2x3/2
=
O.5(x +
3)3/2
x
Lesson 5.4• Applications of Exponential and Power Equations
Name
Period
1. Solve each equation for positive values of x. If answers are not exact,
approximate to the nearest hundredth.
a,
=
d.
4(x5/6
2.6
+ 7)
b. x—’
1
=
159
e. 224
=
=
0.2
5
c. 0.75x
200 (i +
—
8
=
f. 1500 (1 +
—3
=
1525
2. Rewrite each expression in the form ax”.
a. (8x9)2’3
b. (81x12)3”4
c.
d. (—27x—9)’
e.
f• (_ 125X’5)’
(ioo,000xbo)3/5
(49x_b0)2
3. Give the average annual rate of inflation for each situation described.
Give your answers to the nearest tenth of a percent.
a. The cost of a movie ticket increased from $6.00 to $8.50 over
10 years.
b. The monthly rent for Hector’s apartment increased from $650 to
$757 over 4 years.
4. The population of a small town has been declining because jobs have
been leaving the area. The population was 23,000 in 2002 and 18,750
in 2007. Assume that the population is decreasing exponentially.
a. Define variables and write an equation that models the population
in this town in a particular year.
b. Use your model to predict the population in 2010.
c. According to your model, in what year will the population first fall
below 12,000?
30
CHAPTER 5
Discovering Advanced Algebra More Practice Your Skills
©2010 Key Curriculum Press
LESSON 5.4. Applications of
Exponential
and Power Equations
l.a. x 17.58
d.x65,8Q
2. a. 4x
6
d. 81x’
2
3. a. 3.5%
b.x=625
e.x0.05
c. x 1.46
f. x 0.03
b. 27x
9
c. 7x
5
e. 6
00x
0
l
f. —5x
5
b. 3.9%
4. a. Let t represent the year
and P represent the
population p = 23,000(0.96) t—2002
b. 16,592
c. 2018
Lesson 5.6• Logarithmic Functions
Name
Period
Date
1. Rewrite each logarithmic equation in exponential form using the
definition of logarithm. Then solve for x.
3
a. log
=
d. logx
1
=
x
b. logYi
e. 3
=
c. x
=
log 125
=
4 32
log
f. log
x
20
=
1
2. Find the exact value of each logarithm without using a calculator.
Write answers as integers or fractions in lowest terms.
a. 3
log
8
l
b. 5
log
\
/
c. 3
log
-
2
d. log
e. log
4
8
f. log 1,000,000,000
3. Each graph is a transformation of either y
=
lOX or y
=
logx. Write
the equation for each graph.
a.
b.
c.
y
JHI 4L
I
4. Use the change-of-base property to solve each equation. (Round to
four decimal places.)
a. log
5 120
d. 6X
32
=
729
CHAPTER 5
=
x
b. log
O.9
3
e.
7X
=
=
4.88
x
c.
4X
=
f. 12X
=
99
5.75
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LESSON 5.6
l.a.
c.
Logarithmic Functions
3XJ_;X
—4
b.x’1/i;X
32; x
2.5
d. 101
f. 201
4X
125; x
e. x
3
5
b.
2. a. 4
=
10
x; x
x; x
12
=
20
d. —5
c. —1
e.I
3. a. y
.
=
lOc
=
.(10x+2)
4. a. 2.9746
e. 0.8146
—
3
b. —0.0959
f. 0.7039
b. y
=
log(x
c. 3.3147
—
1) + 3
d. 3.6789
Lesson 57• Properties of Logarithms
Name
Period
Date
1. Use the properties of logarithms to rewrite each expression as a single
logarithm.
a. log 21
—
log 7
b. —4 log 2
c. —2
log 5 + 4 log 5
2. Write each expression as a sum or difference of logarithms (or
constants times logarithms). Simplify the result if possible.
a. log
5
(\C)
3
c. log
b. 4
log
(
V’.
3. Determine whether each equation is true or false.
log 32
a. log 8
b. 5
log 9
52
log
=
log 4
—
c. log
--
=
log
=
g. logv’
d. log4
flog 8
i-log 5
f. log
=
—2 log 6
h. log
3 15
15
=
—
=
5 4.5
log
—2 log
29
35
log
=
1
4. Change the form of each expression below using definitions or
properties of logarithms or exponents. Name each definition or
property you use.
a. logr
—
logs
d. logx’”
g.
()
S
c.
e. (cd)m
f. log,xy
h.
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qa+b
b.
Cm/n
log a x
y
0
log
CHAPTER 5
33
LESSON 5.7 Properties of Logarithms
1. a. log 3
b. log
-
c. log 25
5
2
a
+lo
.a
gb—
.l4lo
og
gc
b
r
4
l
1
s
+
o
.l
lo
+
o
g
g
g
t
c. i-log a + iog b + --logc— logx
3. a. False
b. True
c. False
d. True
e. True
f. True
g. False
h. True
4. a. log; quotient property of logarithms
b. a; definition of negative exponents
c. q’
1
q’; product property of exponents
d. mlog x power property of logarithms
e. cmdm; power of a product property
f. loge x + logo y; product property of logarithms
power of a quotient property
g.
h. Y; definition of rational exponents
i. Iogx; change-of-base property
Lesson 5.8 • Applications of Logarithms
Name
Period
Date
1. Solve each equation. Round answers that are not exact to four decimal
places.
a. 19,683
=
3X
b. 9.5(8x)
=
220
c. 0.405
=
15.6(0.72)x
2. Suppose that you invest $5,000 in a savings account. How long would it
take you to double your money under each of the following conditions?
a. 5% interest compounded annually
b. 3.6% interest compounded monthly
3. The Richter scale rating of the magnitude of an earthquake is given by
(-/-)
the formula log
where 10 is a certain small magnitude used as
a reference point. (Richter scale ratings are given to the nearest tenth.)
a. A devastating earthquake, which measured 7.4 on the Richter scale,
occurred in western Turkey in 1999. Express the magnitude of this
earthquake as a multiple of 10.
b. Another earthquake occurred in 1998, centered in Adana, Turkey.
This earthquake measured 6.3 on the Richter scale. Express the
magnitude of this earthquake as a multiple of I.
c. Compare the magnitudes of the two earthquakes.
4. The population of an animal species introduced into an area sometimes
increases rapidly at first and then more slowly over time. A logarithmic
function models this kind of growth. Suppose that a population of
N deer in an area t months after the deer are introduced is given by
the equation N = 325 log(4t + 2). How long will it take for the deer
population to reach 800? Round to the nearest whole month.
34
CHAPTER 5
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©2010 Key Curriculum Press
LESSON 5.8k Applications of Logarithms
l.a. x —9
b. x
2.a. 14.2yr
—
1.5111
c. x
—
11.1144
b. 19.3 yr
3. a. 25,118,864I
b. 1,995,262I
c. The intensity of the 1999 earthquake was
about 12.6 times as great as that of the
1998 earthquake.
4. 6 years (72 months)