A-27 Answers to Selected Exercises 101. x y 0 –2 4 7 f(x) = log2 [4(x – 3)] f(x) = log2 (x – 3) + 2 x = 3 2 4 103. f(x) = log3 f(x) = log3 (x + 1) – 2 x + 1 9 x y 0 –2 8 x = –1 2 105. inverses 107. inverses 109. not inverses 111. f -11x2 = log 3 x 113. f -11x2 = log 5 1x - 12 115. f -11x2 = 10x 117. (a) −1 −5 3 35 (b) logarithmic Summary Exercises on Inverse, Exponential, and Logarithmic Functions 1. inverses 2. not inverses 3. inverses 4. inverses 5. x y y = x 0 6. x y y = x 0 7. It is not one-to-one. 8. x y y = x 0 9. B 10. D 11. C 12. A 13. The functions in Exercises 9 and 12 are inverses of one another. The functions in Exercises 10 and 11 are inverses of one another. 14. ƒ-11x2 = 7x 15. ƒ-11x2 = 1 3 x + 2; Domains and ranges of both ƒ and ƒ-1 are 1-∞, ∞2. 16. ƒ-11x2 = 33 x 2 - 1; Domains and ranges of both ƒ and ƒ-1 are 1-∞, ∞2. 17. ƒ is not one-to-one. 18. ƒ-11x2 = 5x + 1 2 + 3x ; Domain of ƒ = range of ƒ-1 = A -∞, 5 3B ´A 5 3 , ∞B ; Domain of ƒ-1 = range of ƒ = A -∞, - 2 3B ´A - 2 3 , ∞B 19. ƒ is not one-to-one. 20. ƒ-11x2 = 2x2 + 9, x Ú 0; Domain of ƒ = range of ƒ-1 = 33, ∞2; Domain of ƒ-1 = range of ƒ = 30, ∞2 21. log 1/10 1000 = -3 22. loga c = b 23. log2 3 9 = 4 24. log4 1 8 = - 3 2 25. log2 32 = x 26. log27 81 = 4 3 27. 526 28. 5-36 29. 5-36 30. 5256 31. 5-26 32. E1 3F 33. 10, 12 ´11, ∞2 34. E3 2F 35. 556 36. 52436 37. 516 38. 5-26 39. 516 40. 526 41. 526 42. E1 9F 43. E - 1 3F 44. 1-∞, ∞2 4.4 Exercises 1. increasing 3. ƒ-11x2 = log 6 x 5. natural; common 7. There is no power of 2 that yields a result of 0. 9. log 8 = 0.90308999 11. 12 13. -1 15. 1.7993 17. -2.6576 19. 3.9494 21. 0.1803 23. 3.9494 25. 0.1803 27. The logarithm of the product of two numbers is equal to the sum of the logarithms of the n umbers. 29. 3.2 31. 8.4 33. 2.0 * 10-3 35. 1.6 * 10-5 37. poor fen 39. bog 41. rich fen 43. (a) 2.60031933 (b) 1.60031933 (c) 0.6003193298 (d) The whole number parts will vary, but the decimal parts will be the same. 45. 1.6 47. -2 49. 1 2 51. 3.3322 53. -8.9480 55. 10.1449 57. 2.0200 59. 10.1449 61. 2.0200 63. (a) 20 (b) 30 (c) 50 (d) 60 (e) 3 decibels 65. (a) 3 (b) 6 (c) 8 67. 631,000,000I0 69. 52.3 thousand; We must assume that the model continues to be logarithmic. 71. (a) 2 (b) 2 (c) 2 (d) 1 73. 1 75. between 7°F and 11°F 77. 1.13 billion yr 79. 2.3219 81. -0.2537 83. -1.5850 85. 0.8736 87. 1.9376 89. -1.4125 91. 4v + 1 2 u 93. 3 2 u - 5 2 v 95. (a) 4 (b) 25 (c) 1 e 97. (a) 6 (b) ln 3 (c) ln 9 99. D 101. domain: 1-∞, 02 ´10, ∞2; range: 1-∞, ∞2; symmetric with respect to the y-axis 103. ƒ1x2 = 2 + ln x, so it is the graph of g1x2 = ln x translated up 2 units. 105. ƒ1x2 = ln x - 2, so it is the graph of g1x2 = ln x translated down 2 units. Chapter 4 Quiz [4.1] 1. ƒ-11x2 = x 3 + 6 3 [4.2] 2. 546 3. y x 2 –3 –9 0 f(x) = –3x –3 3 domain: 1-∞, ∞2; range: 1-∞, 02 [4.3] 4. y x 4 0 f(x) = log 4 (x + 2) 2 –2 domain: 1-2, ∞2; range: 1-∞, ∞2 [4.2] 5. (a) $18,563.28 (b) $18,603.03 (c) $18,612.02 (d) $18,616.39 [4.4] 6. (a) 1.5386 (b) 3.5427 [4.3] 7. log6 25 is the exponent to which 6 must be raised in order to obtain 25. 8. (a) 546 (b) 556 (c) E 1 16F [4.4] 9. 1 2 log3 x + log3 y - log3 p - 4 log3 q 10. 7.8137 11. 3.3578 12. 12
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