PART I: Solutions to Odd-Numbered Exercises and Practice Tests mCONTINUED--. [] You should know the definition of the natural logarithmic function. logex = lnx, x>O [] You should know the properties of the natural logarithmic function. (a) In 1 = 0 since e° = 1. (b) In e = 1 since e1 = e. (c) In ex = x since ex = ex. (d) If In x = In y, then x = y. [] You should be able to graph logarithmic functions. Solutions to Odd-Numbered Exercises 1. log,, 64 = 3 ==~ 43 = 64 3. 1og7~=-2 ==~ 7-2=~9 5. log324=’~ ==~ 322/5 =4 7. lnl=O =, e°=l 9. 53= 125 =:~ log5125=3 11. 811/’t=3 ~ logs13=¼ 1 ==~ log6 3~ = -2 13. 6-2 -- 36 15. e3 = 20.0855... =:~ In 20.0855... = 3 17. e2"6 = 13.463... ~ In 13.463... = 2.6 19. log2 16 = log2 24 = 4 21. log16(¼) = 1og161 --Io9164 = 0- Io916161/2 = --½23. loglo 0.01 = loglo 10-2 = -2 25. log7x = 10979 x=9 27. In es = x 29. log6 62 = x 8 ¯ In e = x 2log6 6 = x 8=X 2=X 31. loglo 345 ~ 2.538 33. log~o(~) ~--0,097 35. In(4 + ~"~)~ 1.746 37. ln.v/~ ~- 1.869 39. 6 lOglo 14.8 ~ 7.022 41o 12 In 6.4 ~ 22.276 151 PART I: Solutions to Odd-Numbered Exercises and Practice Tests 45. f(x) = ex, g(x) = In x 43. f(x) = 3x, g(x) = log3 x 2 2 fand g are inverses. Their graphs are reflected about the line y = x. 47. f(x) = log3 x + 2 f and g are inverses. Their graphs are reflected about the line y = x. 49. f(x) = -log3(x + 2) Asymptote: x = -2 Point on graph: (- 1, 0) Matches graph (d). Asymptote: x = 0 Point on graph: (1, 2) Matches graph (c). 51. f(x) = log3(1 ’-- x) Asymptote: x = 1 Point on graph: (0, 0) Matches graph (b). 53. f(x) = log4 x Domain: x > 0 =~ The domain is (0, ~x~). Vertical asymptote: x = 0 x-intercept: (1, 0) y=log4x ==~ 4y=x x Y 1 1 4 2 -1 0 1 1_ 55. h(x) = log4(x- 3) Domain: x - 3 > 0 or (3, ~) Vertical asymptote: x = 3 Intercept: (4, 0) y y 6 4 2. 57, y = -log3 x + 2 Domain: (0, Vertical asymptote: x = 0 x-intercept: -log3 x + 2 = 0 2 = log3 x 32 =X 9=x The x-intercept is (9, 0). y = -loga x + 2 log3 x = 2 - y =:~ 32.-y = x y 6 4 2 x 27 9 3 1 1 y -1 0 1 2 3 4 6 8 10 12 bered Exercises and Practice Tests f(x) =6+log6(x-3) Domain: (3, ~x~) Vertical asymptote: x = 3 y 9: 7 x-intercept: log6(x - 3) = -6 5- 6-6=x- 3 X= 3 +6-6~.3 X 4 9 Y 6 7 5 Domain: -_ > 0 =e, x > 0 5 The domain is (0, ~). x Vertical asymptote: ~ = 0 ==~ x = 0 The vertical asymptote is the y-axis. x-intercept: loglo = 0 - = 10o 5 x x -=1 ~ x=5 5 The x-intercept is (5, 0). 63. f(x) = ln(x- 2) Domain: x-2>0 ==~ x>2 The domain is (2, ~). Vertical asymptote: x - 2 = 0 ==~ x = 2 x-intercept: 0 = ln(x - 2) eO=x-2 3=x The x-intercept is (3, 0). 6 4. 32I.~,~- -1 456789 153 PART 1: Solutions to Odd-Numbered Exercises and Practice Tests g(x)= ln(-x) Domain: -x > 0 =e, x < 0 The domain is (-c~, 0). Vertical asymptote: -x = 0 ==~ x = 0 x-intercept: 0 = ln(-x) e0 m --x t -lO The x-intercept is (- 1, 0). 67. f(x) = : x 69. h(x) = 4xlnx (a) (a) Iiiii -/ tii "11 .(b) Domain: (0, o c) (c) Increasing on (2, oo) Decreasing on (0, 2) (b) Domain: (0, ~) (c) Increasing on (0.368, ~) (d) Relative minimum: (2, 1.693) (d) Relative minimum: (0.368, - 1.472) 71. t = Decreasing on (0, 0.368) 10 In 2 ~ 23.68 years In 67 - In 50 73. (a) f(0) = 80 - 17 loglo(0 + 1) = 80 (b) f(4) = 80- 17 loglo(4 + 1)= 68.1 (c) f(10) = 80 - 17 log10(10 + 1) ~- 62.3 (d) ,oo O~ ~ 12 0 75. (a) r 0.005 0.010 0.015 t 138.6 yr 69.3 yr 46.2 yr r 0.020 0.025 0.030 t 34.7 yr 27.7 yr 23.1 yr 11o (b) T > 300°F when p > 67.3 pounds per square inch [The graph of T and y = 300 intersect at p = 67.3.] (c) T(74)= 306.48°F The doubling time decreases as r increases. 154 PART I: Solutions to Odd-Numbered Exercises and Practice Tests (a) I= 1:/3= 101ogl0(l~_t~)= 10. loglo(1012) = 10(12) = 120decibels ’ . / lO-~ (b) I = 10-2:/3 = 101oglo~) = I0 loglo(lOI°) = I0(I0)= I00 decibels (c) No, this is a logarithmic scale. 81. y = 80.4 - 11 lnx, 100_<x< 1500 450 cubic ft per minute (a) = 15 cubic feet per minute per child 30 children (b) From the graph, for y = 15 you get x ~ 382 cubic feet. (c) If ceiling height is 30, then 382 square feet of floor space is needed. . [ 1659.24 ~ 83, t = 16.625 Jn[.165~-~~ : :/50] ~ 10 years 85. Total amount = (1659.24)(10)(12) ~ $199,108.80 Interest = 199,108.80- 150,000 = $49,108.80 87. True. 1og3(27) = log3 33 = 3 91. (a) x 1 0 ~(x) 5 10 89. lO~ 0.322 0.230 0.046 lO~ 106 0.00092 0.0000138 4 (b) As x increases without bound, f(x) approaches O. (c) o.~ 93. Vertical asymptote: x = 0 No horizontal asymptote 95. Vertical asymptote: x = 7 No horizontal asymptote 97. e7/2 ~ 33.115 99. 6e-s ~- 0.002
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