Section 8.5
Logarithmic Functions
829
8.5 Exercises
1
In Exercises 1-18, find the exact value
of the function at the given value b.
√
1. f (x) = log3 (x); b = 5 3.
In Exercises 19-26, use a calculator to
evaluate the function at the given value
p. Round your answer to the nearest
hundredth.
2.
f (x) = log5 (x); b = 3125.
19.
f (x) = ln(x); p = 10.06.
3.
f (x) = log2 (x); b =
20.
f (x) = ln(x); p = 9.87.
4.
f (x) = log2 (x); b = 4.
21.
f (x) = ln(x); p = 2.40.
5.
f (x) = log5 (x); b = 5.
22.
f (x) = ln(x); p = 9.30.
6.
f (x) = log2 (x); b = 8.
23.
f (x) = log(x); p = 7.68.
7.
f (x) = log2 (x); b = 32.
24.
f (x) = log(x); p = 652.22.
8.
f (x) = log4 (x); b =
1
.
16
25.
f (x) = log(x); p = 6.47.
26.
f (x) = log(x); p = 86.19.
9.
1
f (x) = log5 (x); b =
.
3125
1
.
16
10.
f (x) = log5 (x); b =
11.
f (x) = log5 (x); b =
12.
f (x) = log3 (x); b =
13.
f (x) = log6 (x); b =
14.
f (x) = log5 (x); b =
15.
f (x) = log2 (x); b =
16.
1
.
25
√
6
5.
√
3
3.
√
6
6.
√
5
5.
√
6
2.
In Exercises 27-34, solve the given equation, and round your answer to the nearest hundredth.
27.
13 = e8x
28.
2 = 8ex
29.
19 = 108x
30.
17 = 102x
31.
7 = 6(10)x
1
f (x) = log4 (x); b = .
4
32.
7 = e9x
17.
1
f (x) = log3 (x); b = .
9
33.
13 = 8ex
18.
f (x) = log4 (x); b = 64.
34.
5 = 7(10)x
Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/
Version: Fall 2007
830
Chapter 8
Exponential and Logarithmic Functions
In Exercises 35-42, the graph of a logarithmic function of the form f (x) =
logb (x − a) is shown. The dashed red
line is a vertical asymptote. Determine
the domain of the function. Express your
answer in interval notation.
38.
y
5
5
x
35.
y
5
39.
5
x
y
5
5
x
36.
y
5
40.
5
x
y
5
5
x
37.
y
5
41.
5
x
y
5
5
Version: Fall 2007
x
Section 8.5
Logarithmic Functions
831
42.
y
5
5
x
Version: Fall 2007
Chapter 8
Exponential and Logarithmic Functions
8.5 Solutions
1.
√
√
1
1
f ( 5 3) = log3 ( 5 3) = log3 (3 5 ) = .
5
3.
f
1
16
= log2
1
16
1
24
4 !
1
2
2−4
= log2
= log2
= log2
= −4
5.
f (5) = log5 (5) = log5 (51 ) = 1.
7.
f (32) = log2 (32) = log2 (25 ) = 5.
9.
f
1
3125
1
= log5
3125
1
= log5
55
5 !
1
= log5
5
= log5 5−5
= −5
11.
√
√
1
1
f ( 6 5) = log5 ( 6 5) = log5 (5 6 ) = .
6
13.
√
√
1
1
f ( 6 6) = log6 ( 6 6) = log6 (6 6 ) = .
6
15.
√
√
1
1
f ( 6 2) = log2 ( 6 2) = log2 (2 6 ) = .
6
17.
Version: Fall 2007
Section 8.5
Logarithmic Functions
1
1
= log3
f
9
9
1
= log3
32
2 !
1
= log3
3
= log3 3−2
= −2
19.
Using a calculator, f (10.06) = ln(10.06) ≈ 2.31.
21.
Using a calculator, f (2.40) = ln(2.40) ≈ 0.88.
23.
Using a calculator, f (7.68) = log(7.68) ≈ 0.89.
25.
Using a calculator, f (6.47) = log(6.47) ≈ 0.81.
27.
13 = e8x =⇒ ln(13) = ln(e8x )
=⇒ ln(13) = 8x
=⇒ x =
ln(13)
≈ 0.320618669682692
8
29.
19 = 108x =⇒ log(19) = log(108x )
=⇒ log(19) = 8x
=⇒ x =
log(19)
≈ 0.159844200119104
8
31.
7
= 10x
6
7
=⇒ log
= log(10x )
6
7
=x
=⇒ log
6
7
=⇒ x = log
≈ 0.0669467896306132
6
7 = 6(10)x =⇒
Version: Fall 2007
Chapter 8
Exponential and Logarithmic Functions
33.
13
= ex
8
13
=⇒ ln
= ln(ex )
8
13
=⇒ ln
=x
8
13
=⇒ x = ln
≈ 0.485507815781701
8
13 = 8ex =⇒
35. Project all points on the graph onto the x-axis. This is shaded in red in the figure
below. Thus, the domain is the set of all real numbers greater than 0. In interval
notation, the domain equals (0, ∞).
y
5
5
x
37. Project all points on the graph onto the x-axis. This is shaded in red in the figure
below. Thus, the domain is the set of all real numbers greater than −1. In interval
notation, the domain equals (−1, ∞).
y
5
5
Version: Fall 2007
x
Section 8.5
Logarithmic Functions
39. Project all points on the graph onto the x-axis. This is shaded in red in the figure
below. Thus, the domain is the set of all real numbers greater than 0. In interval
notation, the domain equals (0, ∞).
y
5
5
x
41. Project all points on the graph onto the x-axis. This is shaded in red in the figure
below. Thus, the domain is the set of all real numbers greater than −3. In interval
notation, the domain equals (−3, ∞).
y
5
5
x
Version: Fall 2007