Topic 7
Review
TOPIC VOCABULARY
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Check Your Understanding
Fill in the blanks.
1. There are two types of exponential functions. For ? ,
as the value of x increases, the value of y decreases,
approaching zero. For ? , as the value of x increases,
the value of y increases.
2. As x or y increases in absolute value, the graph may
approach a(n) ? .
3. A(n) ? with a base e is a ? .
(
4. The value of 1 + 1x
infinity.
)x approaches
? as x approaches
5. The inverse of an exponential function with a base e is
the ? .
7-1 Attributes of Exponential Functions
Quick Review
Exercises
The general form of an exponential function is y =
where x is a real number, a ≠ 0, b 7 0, and b ≠ 1. When
b 7 1, the function models exponential growth, and
b is the growth factor. When 0 6 b 6 1, the function
models exponential decay, and b is the decay factor. The
y-intercept is (0, a).
abx ,
Determine whether each function is an example of
exponential growth or decay. Then, find the y-intercept.
6. y = 5x
7. y = 2(4)x
8. y = 0.2(3.8)x
9. y = 3(0.25)x
Example
()
1 x
12. y = 2.25 ( 3 )
Determine whether y = 2(1.4)x is an example of
exponential growth or decay. Then, find the y-intercept.
Write a function for each situation. Then find the value of
each function after five years. Round to the nearest dollar.
Since b = 1.4 7 1, the function represents exponential
growth.
14. A $12,500 car depreciates 9% each year.
Since a = 2, the y-intercept is (0, 2).
25 7 x
10. y = 7 5
11. y = 0.0015(10)x
(1)
13. y = 0.5 4
x
15. A baseball card bought for $50 increases 3% in value
each year.
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7-2 Transformations of Exponential Functions
Quick Review
Exercises
You can graph an exponential function by considering
transformations of the parent function.
Graph each function on the same set of axes as the parent
function y = 2x .
In y = 13 10 x , each y-value of the function is 13 the
corresponding y-value of the parent function y = 10 x .
Therefore, the graph of y = 13 10 x is a compression of the
graph of the parent function by a factor of 13 .
16. y = 2x - 7
17. y = 0.25
18. y = 2(x+1)
19. y = - 12
#
#
Graph y =
- 1.
The -1 in y = 2 x - 1 shifts
the graph of the parent
function y = 2x down by
1 unit.
4
#
−4
20. y = - 15 10x
22. y = 10(x-4)
y
2
y = 2x
y = 2x − 1
x
2
4
O
−2
# 2x
Graph each function on the same set of axes as the parent
function y = 10x .
Example
2x
# 2x
−2
21. y = 10x - 25
1
23. y = 2
# 10x + 20
24. Without actually graphing, explain how the graph
of y = -10(x-18) is related to the graph of the parent
function y = 10x .
25. Give examples of two different functions whose parent
function is y = 10x and whose graph passes through the
point (1, 6).
−4
7-3 Attributes and Transformations of f (x) = e x
Quick Review
Exercises
You can graph a natural base exponential function by
considering transformations of the parent function y = e x .
Graph each function on the same set of axes as the parent
function y = e x .
In y = e x - 7, each y-value of the function is 7 less than
the corresponding y-value of the parent function y = e x .
Therefore, the graph of y = e x - 7 is a shift of 7 units down
of the graph of the parent function.
1
26. y = - 4 e x
28. y = e(x+2)
#
The graph of y = e x curves
upward, so the minimum and
maximum values occur at the
endpoints of the interval.
Minimum value: f (0) = e0 = 1
y
Topic 7
Review
# e(x-2)
30. [ -2, 0]
31. [ -1, 1.5]
32. [1, 4]
33. [3, 5]
6
34. Without actually graphing, explain how the graph of
y = e(x-13) + 7.5 is related to the graph of the parent
function y = ex .
4
2
-4
Maximum value: f (2) = e2 ≈ 7.389
330
1
29. y = 2
Analyze the minimum and maximum value of the
function f (x) = e x on the given interval.
Example
Analyze the minimum and
maximum of the function
y = e x on the interval [0, 2].
27. y = e x + 5
-2
O
y = ex
x
2
4
35. A classmate states that the functions y = 2ex , y = 2e(x-6) ,
and y = 2e(x+6) all have the same asymptote. Do you
agree with your classmate? Explain why or why not.
7-4 Exponential Models in Recursive Form
Quick Review
Exercises
You can write a recursive formula for a sequence by
specifying the first term and then writing a rule to relate
each subsequent term of the sequence to the term before it.
36. The table shows the number of books at a library’s
donation center over a period of several months. Write
an explicit formula and a recursive formula to model
the data.
In the sequence 5, 10, 20, 40, 80, . . . the first term is 5. Each
subsequent term is 2 times the term before it.
Month
Books
The recursive formula is a1 = 5 and an = 2an-1 .
Example
The table shows the number of
times a celebrity’s name is
mentioned each week on a media
Web site. Write a recursive formula
to model the data.
Write the first few terms of the
sequence and look for a pattern.
Week
Mentions
1
8
2
12
3
18
4
27
1
2
3
4
54
216
864
3456
37. The function y = 450(1.06)x models the value y, in
dollars, of an antique piece of furniture x years after it
is purchased. Write a recursive formula to model the
situation.
38. You invest $750 in a savings account that pays 1.5%
annual interest. Write an explicit formula and a
recursive formula to model the situation.
#
a2 = 12 = 1.5 8 = 1.5a1
a1 = 8
a3 = 18 = 1.5 12 = 1.5a2
#
The recursive formula is a1 = 8 and an = 1.5an-1 .
7-5 Attributes of Logarithmic Functions
Quick Review
Exercises
If x =
log b x = y. The logarithmic function is
the inverse of the exponential function, so the graphs
of the functions are reflections of one another across
the line y = x. Logarithmic functions can be translated,
stretched, compressed, and reflected, as represented by
y = a log b (x - h) + k, similarly to exponential functions.
Write each equation in logarithmic form.
by , then
When b = 10, the logarithm is called a common
logarithm, which you can write as log x.
Example
Write
5-2
If y =
bx , then
= 0.04 in logarithmic form.
log b y = x.
y = 0.04, b = 5 and x = -2.
So, log 5 0.04 = -2.
39. 62 = 36
40. 2-3 = 0.125
Evaluate each logarithm.
41. log 2 64
1
42. log 3 9
Graph each logarithmic function.
43. y = log 3 x
44. y = log x + 2
45. y = 3 log 2 (x)
46. y = log 5 (x + 1)
How does the graph of each function compare to the
graph of the parent function?
47. y = 3 log 4 (x + 1)
48. y = -log 2 x + 2
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7-6 Properties of Logarithms
Quick Review
Exercises
For any positive numbers, m, n, and b where b ≠ 1, each of
the following statements is true. Each can be used to rewrite
a logarithmic expression.
r log b mn = log b m + log b n,
by the Product Property
m
r log b n = log b m - log b n,
by the Quotient Property
r log b mn = n log b m,
by the Power Property
Write each expression as a single logarithm. Identify any
properties used.
49. log 8 + log 3
50. log 2 5 - log 2 3
51. 4 log 3 x + log 3 7
52. log x - log y
53. log 5 - 2 log x
54. 3 log 4 x + 2 log 4 x
Expand each logarithm. State the properties of logarithms
used.
55. log 4 x2y 3
Example
56. log 4s4t
2
Write 2 log 2 y + log 2 x as a single logarithm. Identify any
properties used.
2 log 2 y + log 2 x
= log 2 y 2 + log 2 x
= log 2 xy 2
Power Property
Product Property
57. log 3 x
58. log (x + 3)2
59. log 2 (2y - 4)3
60. log 5
z2
Use the Change of Base Formula to evaluate each
expression.
61. log 2 7
62. log 3 10
7-7 Transformations of Logarithmic Functions
Quick Review
Exercises
The simplest example of a logarithmic function is the
logarithmic parent function, written f (x) = log b (x), where
b is a positive real number, b ≠ 1. Transformations can
stretch, compress, reflect, or translate the parent function.
Explain the effect of each transformation on the parent
function.
Example
What is the effect of the following transformations on the
parent function y = log b (x)?
y=a
# logb (x)
63. y = 3 log 2 (x)
64. y = log 10 (x) + 4
65. y = log 10 (x + 14)
66. y = -0.5 log b (x) - 1
67. y = log 2 (x - 6) - 3
68. y = 1.3 log b (x + 2.1)
69. Write an equation that defines each logarithmic
function shown in the graph. One of the functions is a
logarithmic parent function.
r The graph will stretch if 0 a 0 7 1.
r The graph will compress, or shrink, if 0 a 0 6 1.
r The graph will reflect across the x-axis if a 6 0.
4
A
2
y = log b (x - c) + d
r The graph will translate
r The graph will translate
r The graph will translate
r The graph will translate
332
Topic 7
Review
0 c 0 units to the left if c 6 0.
0 c 0 units to the right if c 7 0.
0 d 0 units up if d 7 0.
0 d 0 units down if d 6 0.
−2
O
−2
y
B
C
2
4
6
8
x
10
7-8 Attributes and Transformations of the Natural Logarithm Function
Quick Review
Exercises
The inverse of the function y = e x , can be written as either
y = log e x or y = ln x.
Find the domain, range, and asymptote of each function.
The same relationship can be written using the natural
logarithm function or an exponential function of e.
y = ex
Exponential function
x = log e y
Natural logarithmic function
Example
What is the domain, range, x-intercept, and asymptote of
the function f (x) = log e (x + 5)?
The domain is the set of values of x for which the function is
defined, which is x 7 -5.
The range is all real numbers, because the function could
equal any real number.
70. y = log e x + 9
71. y = log e (x - 3.5)
72. f (x) = -log e x - 1
73. f (x) = 3 log e (x + 1)
Find the minimum and maximum values of f (x) = log e x
on the given interval, if each value exists.
74. [1, e5]
75. (0, e -3]
76. (0, 1]
77. [10, 20]
78. Compare the graph of g(x) = 5 log e (x - 2) to the graph
of the parent function, f (x) = log e x
79. The graph of a logarithmic function has an asymptote
at x = -3, and passes through the point ( -2, 4). What
could be the equation for the graph?
The x-intercept is ( -4, 0).
The asymptote is x = -5, the line that the graph approaches
but does not cross.
7-9 Exponential and Logarithmic Equations
Quick Review
Exercises
An equation in the form
= a, where the exponent
includes a variable, is called an exponential equation.
You can solve exponential equations by taking the
logarithm of each side of the equation. An equation that
includes one or more logarithms involving a variable is
called a logarithmic equation.
Solve each equation. Round to the nearest ten-thousandth.
Example
Solve by graphing. Round to the nearest ten-thousandth.
Solve and round to the nearest ten-thousandth.
88. 52x = 20
89. 37x = 160
90. 63x+1 = 215
91. 0.5x = 0.12
bcx
62x
= 75
log 62x = log 75
2x log 6 = log 75
Take the logarithm of both sides.
Power Property of Logarithms
log 75
x = 2 log 6
Divide both sides by 2 log 6.
x ≈ 1.2048
Evaluate using a calculator.
80. 252x = 125
81. 3x = 36
82. 7x-3 = 25
83. 5x + 3 = 12
84. log 3x = 1
85. log 2 4x = 5
86. log x = log 2x2 - 2
87. 2 log 3 x = 54
92. A culture of 10 bacteria is started, and the number of
bacteria will double every hour. In about how many
hours will there be 3,000,000 bacteria?
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7-10 Natural Logarithms
Quick Review
Exercises
The inverse of y = e x is the natural logarithmic function
y = log e x = ln x. You solve natural logarithmic equations in
the same way as common logarithmic equations.
Solve each equation. Check your answers.
Example
95. 2 ln x + 3 ln 2 = 5
Use natural logarithms to solve ln x - ln 2 = 3.
96. ln 4 - ln x = 2
ln x - ln 2 = 3
x
ln 2 = 3
x
3
2=e
x
2 ≈ 20.0855
x ≈ 40.171
334
Topic 7 5HYLHZ
93. e3x = 12
94. ln x + ln (x + 1) = 2
97. 4e (x-1) = 64
Quotient Property
98. 3 ln x + ln 5 = 7
Rewrite in exponential form.
Use a calculator to find e3.
Simplify.
99. An initial investment of $350 is worth $429.20 after six
years of continuous compounding. Find the annual
interest rate.