Intermediate Algebra
Chapter 12 Review
Set up a Table of Coordinates and graph the given functions. Find the
y-intercept. Label at least three points on the graph. Your graph must
have the correct shape. For each exponential function, give the
equation of the horizontal asymptote.
1. f ( x) = 2 x
 1
2. g( x) =  
 3
x
€
3. f(x) = 1+ 2 x
€
4. f ( x) = log2 x
€
5. g( x) = log 1 x
2
€
6. g( x) = 3
€
7. Given f(x) = x and g(x) = x 2 − 9 , find each of the following:
a. f(9) + g(−1)
b. f(x) + g(x)
c. f(25)• g(x)
€
d. (f o g)(5)
e. (g o f)(16)
€
x+2
€
8. Find the equation of the inverse of the given one-to-one functions:
€
x+2
a. f(x) =
€
5
b. f(x) = (x − 4)3
c. f(x) = 3 2x + 1
€
9. Write each of the following as an exponential equation.
1
a. log32 2 =
5
b. log3 81= 4
1
= −2
c. log5
25
€
10. Write each of the following as a logarithmic equation.
€
1
a. 3−4 =
81
€
1/ 2
b. 25 = 5
€11. Graph the inverse of the given one-to-one function on the same
set of axes. Label at least three points on the graph.
€
12. What test do you use to determine if a given graph represents a
function that has an inverse function? State the test.
13. Determine whether each graph represents a one-to one function
(If so, it the function has an inverse function). State your conclusion.
a.
b.
c.
€
€
€
Find the value of each of the following without using a calculator.
Leave your answers in exact form (no logarithms in any answer).
14. log5 1
15. 6log6 5
16. log25 5
1
17. log 4
16
18. log7 7
19. log1000
€
€
€
20. log5 56
21. lne
22. log3 81
€
€
23. log 10
24. eln300
25. log2 32
€
€
€
€
€
€
€
€
€
26. 10
log( 8x)
Solve each equation algebraically without using a calculator. Give
exact answers.
27. log 4 x = 3
28. log x 16 = 4
29. log9 3 = x
1
=x
30. log3
81
31. State the product rule, quotient rule, and power rule for
logarithms. (See section 5, chapter 12.)
Write each expression as a sum or difference of logarithms. Don’t
use a calculator; but, where possible, evaluate logarithm expressions.
(Example: If the answer contains log5 25 , simplify to obtain 2).
32
32. log2
x
5
€
33. ln 2
e
€
€
34. log5
€
€
€
€
€
€
€
€
€
€
€
€
€
€
€
€
€
25
y3
x2
35. logb
z+1
36. log3 x 2 (2x − 1)
Write as a single logarithm. Don’t use a calculator; but, where
possible, evaluate logarithm expressions without a calculator.
37. 3logb x + 2logb y
1
38. ln x + 2ln y
5
39. 6logb ( x + 1) − 3logb y
1
40. 3log2 x + log2 ( y + 1) − log2 y
2
Solve each equation algebraically. Do not use a calculator, and give
all answers in exact form (no logarithms allowed).
41. 4 x = 16
42. 4 x = 32
1
43. 4 x =
4
x
44. 25 = 5
45. 62x+1 = 36
Use a calculator to approximate the following to four decimal places.
(Hint: The change-of-base formula may be needed on a few of
these.)
46. e−1.25
47. log2 17
48. log0.5 2.1
49. log5.1
50. ln4.8
€
€
€
€
€
€
€
€
€
€
€
€
Solve each equation by using an appropriate method. After you have
gotten an exact answer, use your calculator to approximate the
answer to the nearest thousandth (three decimal places).
51. 5 x = 3
52. 10 x = 8.7
53. e2x = 6.1
54. 20 − 2.1x = 0
55. 103x−1 = 3.7
56. 4 x+1 = 5 x
57. log2 (3x + 1) = 7
58. log(2x − 3) = 3.2
59. ln x = −3
60. log5 x − log5 (4x − 1) = 1
61. log 4 ( x + 2) − log 4 (x − 1) = 1
62. ln x + 4 = 1
Solve each of the following application problems algebraically. For
each problem, you may use your calculator but must show enough
work to outline your solution strategy.
63. How much will an investment of $15,000 be worth in 30 years if
the annual interest rate is 5.5% and compounding is
a. quarterly?
b. monthly?
c. continuously?
64. The equation y = 158.97(1.012)x models the population of the
United States from 1950 through 2003. In this equation, y is the
population in millions and x represents the number of years after
1950. Use the equation to estimate the population of the United
States in €
1975. (Use a calculator and round your answer to two
decimal places).
Answers:
1-6. See graphs on following pages.
€
€
7a. –5
7b. x + x 2 − 9
7c. 5x 2 − 45
7d. 4
7e. 7
8a. f −1(x) = 5x − 2
€
€
8b. f −1(x) = 3 x + 4
x3 − 1
−1
8c. f (x) =
€
2
1/ 5
9a. 32 = 2
€
9b. 3 4 = 81
1
€
9c. 5−2 =
€
25
1
€
= −4
10a. log3
81
1
€
10b. log25 5 =
2
11. See graph on following pages.
€
€
12. Use the horizontal line test.
HLT: A function f is one-to-one and
€ thus has an inverse function if there
is no horizontal line that intersects
the graph of the function f at more
than one point.
13a. Yes, the function has an
inverse.
13b. No, the function does not
have an inverse because it does
not pass the horizontal line test.
13c. Yes, the function has an
inverse—it passes the horizontal
line test.
14. 0
15. 5
1
2
17. −2
16.
18.
1
2
19. 3
20. 6
21. 1
22. 4
23.
1
2
24. 300
25. 5
26. 8x
27. x = 64
28. x = 2
29. x = 1/2
30. −4
31. See chapter 12, section5.
32. 5 − log2 x
33. ln5 − 2
€
€
€
€
€
34. 2 − 3log5 y
35. 2logb x − logb (z + 1)
36. 2log3 x + log3 (2x − 1)
(
37. logb x 3 y 2
(
38. ln y 2 5 x
)
(x + 1)
)
39. logb
€
 x 3 y + 1

40. log2 

y


41. x = 2
€
€
€
€
€
y
57.
€
3
5
2
43. x = −1
42. x =
1
2
1
45. x =
2
46. 0.2865
47. 4.0875
48. −1.0704
49. 0.7076
44. x =
€
6
€
50. 1.5686
log3
51. x =
≈ 0.6826
log5
log8.7 log8.7
=
52.x =
≈ 0.9395
log10
1
€ 53. x = log6.1≈ 0.9041
2loge
€ 54. x = log20 ≈ 4.0377
log2.1
log3.7
€
1+
log10
55. x =
≈ 0.5227
3
€
−log4
56. x =
≈ 6.2126
log4 − log5
58.
€59.
€
60.
€61.
27 − 1
x=
≈ 42.3333
3
103.2 + 3
=793.9466
x=
2
x = e−3 ≈ 0.0498
5
x = ≈ 0.2632
19
x=2
2
€62. x = e − 4 ≈ 3.3891
63a. $77,231.65
€63b. $77,810.82
63c. $78, 104.70
64. 214.20 million people
1. f ( x) = 2 x , Horizontal Asymptote is the x-axis or y = 0 (y = 0 is the
€
€
equation of the x-axis)
Y-intercept is (1, 0)
x
y
-1 1/2
0
1
1
2
 1 x
2. g( x) =   , H.A. y = 0 (y = 0 is the equation of the x-axis)
 3
Y-intercept is (1, 0)
x
y
-1 3
0
1
1
1/3
€
3. f(x) = 1+ 2 x . H.A. y = 1. Y-intercept is (0. 2)
x
y
-1 3/2
0
2
1
3
4. f ( x) = log2 x , There is no y-intercept. The graph does not cross the
€
y-axis.
x
y
1/2 -1
1
0
2
1
5. g( x) = log 1 x . Note that the graph will not cross the y-axis.
X
€
2
y
2
-1
1
0
1/2 1
6. g(x) = 3 x+2 . Shift left two units. Horizontal asymptote does not
move—it remains y = 0. Y-intercept is (0, 9).
x g (x)
-3 1/3
€
-2 1
-1 3
11.