Precalculus
Problem Packet 4.3
Logarithmic Functions
4.3 Logarithmic Functions
1. Express in exponential form: log2 32 = 5
2. Express in exponential form: log5 1 = 0
3. Express in exponential form: log4
1
= −4
16
4. Express in exponential form: log8 4 =
2
3
5. Express in exponential form: log 0.1 = −1
6. Express in exponential form: ln 5 = x
7. Express in exponential form: ln(x + 1) = 2
1
8. Express in logarithmic form: 23 = 8
9. Express in logarithmic form: 10−3 = 0.001
10. Express in logarithmic form: 811/2 = 9
11. Express in logarithmic form: 4−3/2 = 0.125
12. Express in logarithmic form: 8−1 =
1
8
13. Express in logarithmic form: ex = 2
14. Express in logarithmic form: ex+1 = .5
2
15. (a) log5 54
(b) log3 1
16. (a) log7 49
(b) log2 32
17. (a) log3
1
27
18. (a) log5 125
19. (a) 2log2 37
(b) log
√
(c) log4 64
(c) log3 32
10
(c) log5 0.2
(b) log49 7
(c) log9
(b) 10log 5
(c) eln
3
√
5
√
3
20. (a) log8 0.25
21. (a) ln e4
(b) log4
(b) ln
√
2
1
e
22. Solve for x: log2 x = 5
23. Solve for x: log2 16 = x
24. Solve for x: logx 1000 = 3
25. Solve for x: log 0.1 = x
4
(c) log4
1
2
26. Solve for x: logx 25 = 2
27. Solve for x: logx 16 = 4
28. Solve for x: logx 6 =
1
2
29. Solve for x: logx 8 =
3
2
5
30. Find the function of the form y = loga x whose graph is given below.
31. Find the function of the form y = loga x whose graph is given below.
32. Find the function of the form y = loga x whose graph is given below.
6
33. Find the function of the form y = loga x whose graph is given below.
Graph the following functions, not by plotting points, but by applying transformations. State the
domain, range and asymptotes.
34. f (x) = log2 (x − 4)
7
35. f (x) = log5 (−x)
36. f (x) = − log x
37. f (x) = 2 + log3 x
8
38. f (x) = log3 (x − 1) − 2
39. f (x) = 1 − log x
40. f (x) = 3 + ln x
9
41. Find the domain of f (x) = log(2 + 5x)
42. Find the domain of f (x) = log2 (10 − 3x)
43. Find the domain of f (x) = log3 (x2 − 1)
44. Find the domain of f (x) = ln x + ln(2 − x)
45. Find the domain of f (x) =
√
x − 2 − log5 (10 − x)
10
ANSWERS
1. 25 = 32
2. 50 = 1
3. 2−4 =
1
16
4. 82/3 = 4
5. 10−1 = 0.1
6. ex = 5
7. e2 = x + 1
8. log2 8 = 3
9. log 0.001 = −3
10. log81 9 =
1
2
11. log4 0.125 = −
3
2
1
= −1
8
13. ln 2 = x
12. log8
14. ln 0.5 = x + 1
15. (a) 4
(b) 0
(c) 3
16. (a) 2
(b) 5
(c) 2
17. (a) − 3
18. (a) 3
(b)
19. (a) 37
20. (a) −
(b)
1
2
1
2
(b) 5
2
3
21. (a) 4
1
4
(b) − 1
(b)
(c) − 1
1
4
√
(c) 5
(c)
(c) −
1
2
22. x = 32
23. x = 4
24. x = 10
25. x = −1
26. x = 5
27. x = 2
11
28. x = 36
29. x = 4
30. y = log5 x
31. y = log2 x
32. y = log9 x
33. y = log3 x
34. Domain: (4, ∞)
Range: (−∞, ∞)
Asymptote: x = 4
35. Domain: (−∞, 0)
Range: (−∞, ∞)
Asymptote: x = 0
12
36. Domain: (0, ∞)
Range: (−∞, ∞)
Asymptote: x = 0
37. Domain: (0, ∞)
Range: (−∞, ∞)
Asymptote: x = 0
13
38. Domain: (−1, ∞)
Range: (−∞, ∞)
Asymptote: x = −1
39. Domain: (0, ∞)
Range: (−∞, ∞)
Asymptote: x = 0
14
40. Domain: (0, ∞)
Range: (−∞, ∞)
Asymptote: x = 0
2
− ,∞
5
10
−∞,
3
41. Domain:
42. Domain:
43. Domain: (−∞, −1) ∪ (1, ∞)
44. Domain: (0, 2)
45. Domain: [2, 10)
15