James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 1.5 Inverse Functions and Logarithms 67
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20. In the theory of relativity, the mass of a particle with speed is
m 0
m − f svd −
s1 2 v 2 yc 2
where m 0 is the rest mass of the particle and c is the speed of
light in a vacuum. Find the inverse function of f and explain
its meaning.
21–26 Find a formula for the inverse of the function.
21. f sxd − 1 1 s2 1 3x 22. f sxd − 4x 2 1
2x 1 3
23. f sxd − e 2x21 24. y − x 2 2 x, x > 1 2
25. y − lnsx 1 3d 26. y − 1 2 e2x
1 1 e 2x
27–28 Find an explicit formula for f 21 and use it to graph f 21 ,
f, and the line y − x on the same screen. To check your work, see
whether the graphs of f and f 21 are reflections about the line.
27. f sxd − s4x 1 3 28. f sxd − 1 1 e 2x
29–30 Use the given graph of f to sketch the graph of f 21 .
29. y
30. y
1
0 1
x
1
0 2
31. Let f sxd − s1 2 x 2 , 0 < x < 1.
(a) Find f 21 . How is it related to f ?
(b) Identify the graph of f and explain your answer to part (a).
32. Let tsxd − s 3 1 2 x 3 .
(a) Find t 21 . How is it related to t?
(b) Graph t. How do you explain your answer to part (a)?
33. (a) How is the logarithmic function y − log b x defined?
(b) What is the domain of this function?
(c) What is the range of this function?
(d) Sketch the general shape of the graph of the function
y − log b x if b . 1.
34. (a) What is the natural logarithm?
(b) What is the common logarithm?
(c) Sketch the graphs of the natural logarithm function and the
natural exponential function with a common set of axes.
35–38 Find the exact value of each expression.
35. (a) log 2 32 (b) log 8 2
1
36. (a) log 5 125 (b) lns1ye 2 d
37. (a) log 10 40 1 log 10 2.5
(b) log 8 60 2 log 8 3 2 log 8 5
x
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38. (a) e 2ln 2 (b) e lnsln e3 d
39–41 Express the given quantity as a single logarithm.
39. ln 10 1 2 ln 5 40. ln b 1 2 ln c 2 3 ln d
41.
1
3 lnsx 1 2d3 1 1 2 fln x 2 lnsx 2 1 3x 1 2d 2 g
42. Use Formula 10 to evaluate each logarithm correct to six
decimal places.
(a) log 5 10 (b) log 3 57
43–44 Use Formula 10 to graph the given functions on a
common screen. How are these graphs related?
43. y − log 1.5 x, y − ln x, y − log 10 x, y − log 50 x
44. y − ln x, y − log 10 x, y − e x , y − 10 x
45. Suppose that the graph of y − log 2 x is drawn on a coordinate
grid where the unit of measurement is an inch. How
many miles to the right of the origin do we have to move
before the height of the curve reaches 3 ft?
46. Compare the functions f sxd − x 0.1 and tsxd − ln x by
graphing both f and t in several viewing rectangles.
When does the graph of f finally surpass the graph of t?
47–48 Make a rough sketch of the graph of each function.
Do not use a calculator. Just use the graphs given in Figures 12
and 13 and, if necessary, the transformations of Section 1.3.
47. (a) y − log 10sx 1 5d (b) y − 2ln x
48. (a) y − lns2xd (b) y − ln | x |
49–50 (a) What are the domain and range of f ?
(b) What is the x-intercept of the graph of f ?
(c) Sketch the graph of f.
49. f sxd − ln x 1 2 50. f sxd − lnsx 2 1d 2 1
51–54 Solve each equation for x.
51. (a) e 724x − 6 (b) lns3x 2 10d − 2
52. (a) lnsx 2 2 1d − 3 (b) e 2x 2 3e x 1 2 − 0
53. (a) 2 x25 − 3 (b) ln x 1 lnsx 2 1d − 1
54. (a) lnsln xd − 1 (b) e ax − Ce bx , where a ± b
55–56 Solve each inequality for x.
55. (a) ln x , 0 (b) e x . 5
56. (a) 1 , e 3x21 , 2 (b) 1 2 2 ln x , 3
57. (a) Find the domain of f sxd − lnse x 2 3d.
(b) Find f 21 and its domain.
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