334 CHAPTER 5 Exponential and Logarithmic Functions 12. Multiple Choice Writing − + x y z log log 2 log a a a as a single logarithm results in which of the following? (a) ( ) − + x y z log 2 a (b) ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ xz y loga 2 (c) ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ xz y log 2 a (d) ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ x yz loga 2 9. True or False ( ) ( ) = x x log 3 4 log 3 2 4 2 10. True or False ( ) = log 2 3 log2 log3 11. Multiple Choice Choose the expression equivalent to 2 .x (a) e x2 (b) ex ln2 (c) e x log2 (d) e x 2 ln Skill Building In Problems 13–28, use properties of logarithms to find the exact value of each expression. Do not use a calculator. 13. log 77 29 14. − log 2 2 13 15. −e ln 4 16. e ln 2 17. 9log 13 9 18. eln8 19. + log 2 log 4 8 8 20. + log 9 log 4 6 6 21. − log 35 log 7 5 5 22. − log 16 log 2 8 8 23. ⋅ log 6 log 8 2 6 24. ⋅ log 8 log 9 3 8 25. − 4log 6 log 5 4 4 26. + 5log 6 log 7 5 5 27. e log 16 e2 28. e log 9 e2 In Problems 29–36, suppose that = a ln2 and = b ln3 . Use properties of logarithms to write each logarithm in terms of a and b. 29. ln 6 30. ln 2 3 31. ln 1.5 32. ln 0.5 33. ln 8 34. ln 27 35. ln 6 5 36. ln 2 3 4 In Problems 37–56, write each expression as a sum and/or difference of logarithms. Express powers as factors. 37. x log 36 6 38. x log 9 3 39. y log5 6 40. x log 7 5 41. ( ) ex ln 42. e x ln 43. x e ln x 44. ( ) xe ln x 45. ( ) > > u v u v log 0, 0 a 2 3 46. ( ) > > a b a b log 0, 0 2 2 47. ( ) − < < x x x ln 1 0 1 2 48. ( ) + > x x x ln 1 0 2 49. ( ) − > x x x log 3 3 2 3 50. + − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ > x x x log 1 1 1 5 3 2 2 51. ( ) ( ) + + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ > x x x x log 2 3 0 2 52. ( ) + − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ > x x x x log 1 2 2 3 2 53. ( ) − − + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ > x x x x ln 2 4 2 2 2 1 3 54. ( ) − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ > x x x ln 4 1 4 2 2 2 3 55. ( ) + − > x x x x ln 5 1 3 4 4 3 56. ( ) − + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ < < x x x x ln 5 1 4 1 0 1 2 3 2 In Problems 57–70, write each expression as a single logarithm. 57. +u v 3 log 4 log 5 5 58. −u v 2 log log 3 3 59. − x x log log 3 3 3 60. ( ) ( ) + x x log 1 log 1 2 2 2 61. ( ) ( ) − − + x x log 1 5 log 1 4 2 4 62. ( ) ( ) + + − + x x x log 3 2 2 log 1 2 63. ( ) ( ) ( ) − + + − − x x x x x ln 1 ln 1 ln 1 2 64. ( ) ( ) + − − − + + + x x x x x x log 2 3 4 log 7 6 2 2 2 2 65. ( ) − − + x x 8 log 3 2 log 4 log 4 2 2 2 66. ( ) + − x x 21 log log 9 log 9 3 3 3 2 3 67. ( ) ( ) − + x x 2 log 5 1 2 log 2 3 a a 3 68. ( ) ( ) + + + x x 1 3 log 1 1 2 log 1 3 2 69. ( ) ( ) ( ) + − + − − x x x 2 log 1 log 3 log 1 2 2 2 70. ( ) ( ) + − − − x x x 3log 3 1 2log 2 1 log 5 5 5 In Problems 71–78, use the Change-of-Base Formula and a calculator to evaluate each logarithm. Round your answer to three decimal places. 71. log 21 3 72. log 18 5 73. log 71 1 3 74. log 15 1 2 75. log 7 2 76. log 85 77. π e log 78. π log 2 In Problems 79–84, graph each function using a graphing utility and the Change-of-Base Formula. 79. = y x log4 80. = y x log5 81. ( ) = + y x log 2 2 82. ( ) = − y x log 3 4 83. ( ) = + − y x log 1 x 1 84. ( ) = − + y x log 2 x 2
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