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C H A P T E R 3Exponential and Logarithmic FunctionsSection 3.1Exponential Functions and Their Graphs .236Section 3.2Logarithmic Functions and Their Graphs .244Section 3.3Properties of Logarithms .252Section 3.4Exponential and Logarithmic Equations .262Section 3.5Exponential and Logarithmic Models .273Review Exercises .287Problem Solving .299Practice Test .303 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
C H A P T E R 3Exponential and Logarithmic FunctionsSection 3.1 Exponential Functions and Their Graphs15. f ( x) 2 x1. algebraic2. transcendentalDecreasingAsymptote: y 03. One-to-One4. natural exponential; naturalr 5. A P 1 n 16. f ( x) 2 x 2IncreasingAsymptote: y 07. f (1.4) (0.9)1.4 0.8638. f ( π ) 4.7 π 0.0089. f() 32510. f ( 103 ) Matches graph (a).nt6. A Pe rt25Intercept: (0, 1) 1.552( 14 )Intercept: 0,Matches graph (b).17. f ( x) 7 x 2 10120.0200.1431749x( 3) ( )5 32 10f ( x) 0.544(11. f ( 1.5) 5000 2 1.5y)5 1767.7674312 2412. f ( 24) 200(1.2) 1.274 1013. f ( x) 2xIncreasingAsymptote: y 0Intercept: (0, 1)Matches graph (d).2251 3 2 1x123 118. f ( x) 7 xxf ( x) 2 101249710.1430.020y14. f ( x) 2 x 15Increasing4Asymptote: y 1Intercept: (0, 2)Matches graph (c).2361 3 2 1x123 1 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Section 3.119. f ( x) ( 14 )f ( x)23722. f ( x) 4 x 1 x 2 10120.0630.251416xExponential Functions and Their Graphs 2 10120.25141664xf ( x)yy55443211 3 2 120. f ( x) 3x12 1( 14 )xf ( x) 2 13x123 123. f ( x) 2 x 1 3x 3 2 1013.253.5457xf ( x) 2 101216410.250.063y7y6554433211 3 2 1x12321. f ( x) 4 x 1f ( x)12324. f ( x) 3x 2 1 1 101231.0371.1111.33324x 2 10120.0160.0630.2514xx 5 4 3 2 1 1f ( x)y7y6554433221 3 2 1 3 2 1 1x1 12x12345325.3x 1 273x 1 33x 1 3x 226.2 x 2 642 x 2 26x 2 6x 8 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
23827.Chapter 3( 12 )( 12 )xxExponential and Logarithmic Functions37. f ( x) 3e x 4 32 ( 12 ) 5x 528.5x 2 x 8 7 6 5 4f ( x)0.0550.1490.4061.1043Asymptote: y 01125y5 x 2 5 38x 2 376x 15429. f ( x) 3x , g ( x) 3x 132Because g ( x) f ( x) 1, the graph of g can beobtained by shifting the graph of f one unit upward.30. f ( x) ( 72 ) , g ( x) ( 72 )x xBecause g ( x) f ( x), the graph of g can be1xf ( x)by reflecting the graph of f in the x-axis and shifting theresulting graph five units upward.33.f ( x) ef (1.9) ex1.9 6.686 101240.1718.96320.4460.100y5obtained by reflecting the graph of f in the y-axis andshifting f three units to the right. (Note: This isequivalent to shifting f three units to the left and thenreflecting the graph in the y-axis.)g ( x) f ( x) 5, so the graph of g can be obtained 2Asymptote: y 0Because g ( x) f ( x 3), the graph of g can be32. f ( x) 0.3x , g ( x) 0.3x 5138. f ( x) 2e 1.5 xobtained by reflecting the graph of f in the x-axis andy-axis (reflect the graph of f in the origin).31. f ( x) 10 x , g ( x) 10 x 3x 8 7 6 5 4 3 2 121 3 2 1x123 139. f ( x) 2e x 2 4x 2 1012f ( x)4.0374.1004.2714.7366Asymptote: y 4y34. f ( x) 1.5e(1 2)x98765 1.5e120 1.956 105235. f (6) 5000e0.06(6) 7166.64732136. f ( x) 250e0.05 x 250e0.05(20) 679.570 3 2 1x1 2 3 4 5 6 7 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Section 3.1Exponential Functions and Their Graphs44. h( x ) e x 240. f ( x) 2 e x 54x02456f ( x)2.0072.0502.36834.718Asymptote: y 2 28e 3 x 2 e345.7640y3x 2 353x 14313x 1 1239x12345678e 2 x 1 e 446.2x 1 441. s(t ) 2e0.5t2x 55x 447.4ex2 3 e2 xx2 3 2x 1x2 2x 3 042. s(t ) 3e 0.2t(x20 3)( x 1) 0x 3or x 148. 16ex172 6 e5 xx2 6 5x 2x2 5x 6 0(x43. g ( x) 1 e x 3)( x 2) 0x 3 or x 24 3523049. P 1500, r 2%, t 10 yearsr Compounded n times per year: A P 1 n nt10 n0.02 1500 1 n Compounded continuously: A Pe rt 1500e 0.02(10)n12412365ContinuousA 1828.49 1830.29 1831.19 1831.80 1832.09 1832.10 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
240Chapter 3Exponential and Logarithmic Functions50. P 2500, r 3.5%, t 10 yearsr Compounded n times per year: A P 1 n nt10 n0.035 2500 1 n Compounded continuously: A Pe rt 2500e 0.035(10)n12412365ContinuousA 3526.50 3536.95 3542.27 3545.86 3547.61 3547.6751. P 2500, r 4%, t 20 yearsr Compounded n times per year: A P 1 n nt0.04 2500 1 n 20 nCompounded continuously: A Pe rt 2500e 0.04(20)n12412365ContinuousA 5477.81 5520.10 5541.79 5556.46 5563.61 5563.8552. P 1000, r 6%, t 40 years0.06 Compounded n times per year: A 1000 1 n 40 nCompounded continuously: A 1000e 0.06(40)n12412365ContinuousA 10,285.72 10,640.89 10,828.46 10,957.45 11,021.00 11,023.1853. A Pe rt 12,000e0.04tt1020304050A 17,901.90 26,706.49 39,841.40 59,436.39 88,668.6754. A Pe rt 12,000e0.06tt1020304050A 21,865.43 39,841.40 72,595.77 132,278.12 241,026.4455. A Pe rt 12,000e0.065tt1020304050A 22,986.49 44,031.56 84,344.25 161,564.86 309,484.0856. A Pe rt 12,000e0.035tt1020304050A 17,028.81 24,165.03 34,291.81 48,662.40 69,055.23 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Section 3.1Exponential Functions and Their Graphs24157. A 30,000e(0.05)(25) 104,710.2958. A 5000e(0.075)(50) 212,605.41C (t ) 29.88(1.04)59.tTen years from today, t 10: C (10) 29.88(1.04)10 44.2360. V (t ) 100e 4.6052t(a) V (1) 10,000 computers(b) V (1.5) 100,004 computers(c) V ( 2) 1,000,060 computers61. (a)42525350(b)55tP (in millions)tP (in millions)tP (in millions)tP (in millions)tP (in millions)tP (in millions)tP (in millions)tP (in .989403.080535455405.182407.294409.417(c) Using the model and extending the table beyond the year 2055, the population will exceed 430 million in 2064.tP (in 98 415.854 418.022 420.202 422.393 424.595 426.808 429.034 431.270 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
242Chapter 3Exponential and Logarithmic Functions62. (a) Because the coefficient of the model P 57.59e 0.0051t is positive, the population is increasing.(b) 2003: Let t 3: P 57.59e 0.0051(3) 58.478The population of Italy in 2000 was 58,478,000 people.2015: Let t 15: P 57.59e 0.0051(15) 62.169The population of Italy in 2012 was about 62,169,000 people.(c) 2020: Let t 20: P 57.59e 0.0051(20) 63.774The population of Italy in 2020 will be about 63,774,000 people.2025: Let t 25: P 57.59e0.0051(25) 65.421The population of Italy in 2025 will be about 65,421,000 people.( 12 )63. Q 1667. True. The line y 2 is a horizontal asymptote for thet 24,100graph of f ( x) 10 x 2. As x , f ( x) 2(a) Q(0) 16 gramsbut never reaches 2.(b) Q(75,000) 1.85 grams(c)68. False, e 20271,801. e is an irrational number.99,99069. f ( x) 3x 20 3x3 2150,0000( 12 )64. Q 10t 5715( 12 )(a) When t 0: Q 100 5715 10(1) 10 grams( 12 )(b) When t 2000: Q 102000 5715 7.85 grams(c)So, f ( x) g ( x), but f ( x) h( x).70. g ( x) 2 2 x 6 2 2 x 26QMass of 14C (in grams) 1 3x 2 3 1 (3 x )9 h( x ) 64( 2 2 x )1210 64( 2 2 )8x 64( 4 x )642t40008000Time (in years)( 78 )65. (a) V (t ) 49,810twhere t is the number of years( 78 )4 29, 197.71After 4 years, the value of the van is about 29,198.66. (a) C (t ) 300(0.75)So, g ( x) h( x) but g ( x) f ( x).t(b) C (8) 30 milligrams per milliliter( ) x 16( 22 ) 16( 2 2 x )71. f ( x) 16( 4 x ) and f ( x) 16 4 x 42 ( 4 x )since it was purchased.(b) V ( 4) 49,810 h( x ) 42 x( 14 ) ( 14 ) ( 2 x) h( x)x 2 g ( x)So, f ( x) g ( x) h( x). 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Section 3.172. f ( x ) e x 3Exponential Functions and Their Graphs75.7g ( x ) e 3 x e3 e x(y1 1 1(h( x ) e x 3 e x e 3)243x(xy2 e 66So, none are equal. 1As x increases, the graph of y1 approaches e, whichis y2 .73. y 3x and y 4 xyx3y 4x0.5 0.576. f ( x ) 1 and g ( x) e (Horizontal line)x y 3x241 2fx 11g2 1 330x–2 1012As x , f ( x) g ( x).3x1913139As x , f ( x) g ( x).4x11614141677. (a)y2 x 2(a) 4 x 3 x when x 0.3y1 2 x(b) 4 x 3 x when x 0. 374. (a) f ( x) x 2e x3 15 2At x 2, both functions have a value of 4. Thefunction y1 increases for all values of x. Thefunction y2 is symmetric with respect to the y-axis.7(b) 1y1 3 xy2 x 33Decreasing: ( , 0), ( 2, )Increasing: (0, 2) 3(Relative maximum: 2, 4e 2)Relative minimum: (0, 0)(b) g ( x) x 23 x 1Both functions are increasing for all values of x. Forx 0, both functions have a similar shape. Thefunction y2 is symmetric with respect to the origin.In both viewing windows, the constant raised to avariable power increases more rapidly than the variableraised to a constant power.6 2310 2Decreasing: (1.44, )Increasing: ( , 1.44)Relative maximum: (1.44, 4.25) 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
244Chapter 3Exponential and Logarithmic Functions78. The graph of y a x is increasing, so graphs (d), (e), and (f) represent y 2x , y e x , and y 10 x.The greater the value of a, the quicker the graph increases. Because 10 e 2, y 10x matches graph (d),y e x matches graph (e), and y 2 x matches graph (f).Graph (c) is a reflection in the y-axis of graph (d), so y 10 x matches graph (c).Graph (b) is a reflection in the y-axis of graph (e), so y e x matches graph (b).Graph (a) is a reflection in the y-axis of graph (f), so y 2 x matches graph (a)79. The functions (c) h ( x ) 3 x and (d) k ( x ) 2 x are exponential.80. P 3000, r 6%, t 10 yearsr A P 1 n nt10 n0.06 3000 1 n (a) n 365A 5466.09(b) n 365( 24) 8760A 5466.35(c) n 365( 24)(60) 525,600A 5466.36(d) n 365( 24)(60)(60) 31,536,000A 5466.38Increasing the number of compoundings does not result in unlimited growth because the balance increases slowly fromcompounding by the day to compounding by the second.Section 3.2 Logarithmic Functions and Their Graphs1. logarithmic13. 4 3 2. 10 3f (64) log 2 64 6 because 2 6 645. x y16. f ( x) log 25 x6. positive real numbersf (5) log 25 5 7. log 4 16 2 42 16 2 9 2 1819. log12 12 1 12 1225 322 5 411. 53 125 log5 125 312. 93 2 27 log 9 27 3212because 251 2 517. f ( x) log8 xf (1) log8 1 0 because 8 0 1110. log32 4 16415. f ( x) log 2 x4. a log a x x181 log 414. 240 1 log 24 1 03. natural; e8. log 916418. f ( x) log xf (10) log 10 1 because 101 1019.g ( x) log a xg ( a 2 ) log a a 2 2 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Section 3.2g ( x) log b x20.g (b 3) log bLogarithmic Functions and Their Graphs33.12b log b b x12f ( x) 7 x21. f ( x) log xf( 5001 ) log 500112149171749149171749 2 1012y 2.699f(x) 7 x54f ( x ) log x23.0g ( x) log7 xf ( x) log xf 1x( 78 ) log( 78 ) 0.05822. 22453f (12.5) log 12.5 1.097g(x) log 7 x21f ( x ) log x24. 1f (96.75) log 96.75 1.98625. log8 8 1 because 81 834.x123x26. logπ π 2 2 because π 2 π 2f ( x) 5 x27. log7.5 1 0 because 7.50 1x29. log5 ( x 1) log 5 6 10121251515251525 2 1012f(x) 5 xg(x) log 5 x130. log 2 ( x 3) log 2 9 2x 3 9x 112 1x 12 2)11 x 2 72x 4x 2log( x 2 6 x) log 27x 2 6 x 2735.x 2 1012136161636f ( x) 6 xxg ( x) log6 xx 2 6 x 27 0(x152x 532.125yx 1 6(5 2g ( x) log5 x28. 5log5 3 3 because aloga x x, 5log5 3 331. log 11 log x 2 74 1136161636 2 1012y 9)( x 3) 0f(x) 6 x4x 9 0 x 93x 3 0 x 3g(x) log 6 x21x 11234 1 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
24636.Chapter 3Exponential and Logarithmic Functions 2 1012f ( x) 10 x1100110110100110110100g ( x) log10 x 2 1012 2 112 1 237. f ( x) log3 x 2Asymptote: x 0Point on graph: (1, 2)14142 10112Matches graph (a).The graph of f ( x) is obtained from g ( x) by shifting thegraph two units upward.38. f ( x) log3 ( x 1)Asymptote: x 1Point on graph: ( 2, 0)y log6 x 6 xPoint on graph: (0, 0)Matches graph (b).1 266120143. y log 3 x 1Domain: (0, )x-intercept:log 3 x 1 0log 3 x 13 1 xf ( x) shifts g ( x) one unit to the right.Asymptote: x 1 1yMatches graph (d).39. f ( x) log3 (1 x) log3 ( x 1) 3x 1Vertical asymptote: x 0 121x-intercept: (1, 0)g ( x)12Domain: (0, )163y42. g ( x) log6 xx2 2g(x) log 10 xx1 1y log 4 x 4 y xf ( x)10 xx 1Vertical asymptote: x 0xy11x-intercept: (1, 0)1100f(x) 2Domain: (0, )x2y41. f ( x) log 4 xx13 xThe x-intercept is( 13 , 0).Vertical asymptote: x 0y log 3 x 1log3 x y 1 3y 1 xThe graph of f ( x) is obtained by reflecting the graphx13039of g ( x) in the y-axis and shifting the graph one unit to19y 10123the right.y40. f ( x) log3 x2Asymptote: x 01Point on graph: (1, 0)Matches graph (c).f ( x) reflects g ( x) in the x-axis.x 1123 1 2 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Section 3.244. h( x) log 4 ( x 3)The domain is (3, ).Domain: x 1 0 x 1The domain is (1, ).yx-intercept:log 5 ( x 1) 4 04240 x 32468 4The x-intercept is ( 4, 0).Vertical asymptote: x 3 0 x 3x3 144719h( x ) 1012 x 1626625 x321x23456,0.( 626625 )y log5 ( x 1) 4 5 y 4 1 xDomain: x 2 0 x 2The domain is ( 2, ).x1.000321.00161.0081.041.2y 10123 x 47. y log 7 yx-intercept:4Domain:0 log 6 ( x 2)20 log 6 ( x 2)x606 x 2 21 x 2 4 1 xThe x-intercept is ( 1, 0).Vertical asymptote: x 2 0 x 2y log 6 ( x 2) y log 6 ( x 2)x 0 x 07The domain is (0, ). x x-intercept: log 7 x7x7xy6 042 100 2 1x4 26810 4 6 7The x-intercept is (7, 0). 2 xf ( x)4Vertical asymptote: x 1 0 x 145. f ( x) log6 ( x 2)x1625The x-intercept isy log 4 ( x 3) 4 y 3 x55 4 x 110 24 x6log 5 ( x 1) 4x1 x 3yx-intercept:6log 4 ( x 3) 0624746. y log5 ( x 1) 4Domain: x 3 0 x 3 yLogarithmic Functions and Their Graphs4 1 156 13536 1012Vertical asymptote:x 0 x 07The vertical asymptote is the y-axis.x 12345y 0.85 0.54 0.37 0.24 0.15x 67 8y 0.069 0 0.06 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
248Chapter 3Exponential and Logarithmic Functions48. y log( 2 x)Domain: 2 x 0 x 0gThe domain is ( , 0).g100 2 x1 2x(( 12 ) ln 12Vertical asymptote: x 063. 2.5 ln 1 2.5(0) 0y log( 2 x) 100 2 x10 y 2 x64. 12 10 y xx1 2001 20 12 5 50 500y 2 10123yln eπ1 π65. ln eln e ln e1 166. eln(1 e) eln e e 1 1eDomain: x 4 0 x 41The domain is ( 4, ).x 11x-intercept: 0 ln ( x 4) 1e0 x 4 249. ln 167. f ( x) ln( x 4)212 0.693 1 62. ln 2 ln e 2 2 e )The x-intercept is 12 , 0 . 25 6.43861. e ln 4 4 12x ( 5 ) 8 ln60. g ( x) ln xx-intercept: log( 2 x) 0 3g ( x) 8 ln x59.5 x 0.693 e 0.693 1250. ln 7 1.945 e1.945 751. ln 250 5.521 e5.521 The x-intercept is (5, 0).Vertical asymptote: x 4 0 x 4x 250f ( x)567 0.6900.691.10y52. ln 1 0 e0 153. e2 7.3890 ln 7.3890 24.542x54. e 3 4 0.4723 ln 0.4723 342468 255. e 4x 12 ln12 4x 456. e2 x 3 ln 3 2 x57. f ( x) ln xf (18.42) ln 18.42 2.91358. f ( x) 3 ln xf (0.74) 3 ln 0.74 0.903 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Section 3.268. h( x) ln( x 5)Logarithmic Functions and Their Graphs70. f ( x) ln(3 x)Domain: x 5 0 x 5Domain: 3 x 0 x 3The domain is ( 5, ).The domain is ( , 3).x-intercept: ln (3 x) 00 ln ( x 5)x-intercept:0e0 3 xe x 5 4 x1 3 x2 xThe x-intercept is ( 4, 0).Vertical asymptote: x 5 0 x 5x 4.5 4 3 2h( x ) 0.6900.691.10The x-intercept is ( 2, 0).Vertical asymptote: 3 x 0 x 3y ln (3 x) 3 e y xx2.952.862.6320.28f ( x) 3 2 101y4y2 6249x 2322 2 4 2 169. g ( x) ln( x)x124 1 2 3Domain: x 0 x 0The domain is ( , 0).71. f ( x) ln( x 1)x-intercept:30 ln( x)e0 x90 1 xThe x-intercept is ( 1, 0). 3Vertical asymptote: x 0 x 0x 0.5 1 2 3g ( x) 0.6900.691.1072. f ( x) ln( x 2)3 45y 3273. f ( x) ln x 81 3 210x 11 2 218 2 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
250Chapter 3Exponential and Logarithmic Functions78. ln ( x 2 2) ln 2374. f ( x) 3 ln x 14x 2 2 23 5x 2 2510x 5x 79. t 16.625 ln , x 750 x 750 675. ln ( x 4) ln 12(a) When x 897.72:x 4 12897.72 t 16.625 ln 30 years 897.72 750 x 876. ln ( x 7) ln 7When x 1659.24:x 7 71659.24 t 16.625 ln 10 years 1659.24750 x 14(b) Total amounts:(897.72)(12)(30) 323,179.20 323,179ln ( x 2 x) ln 677.x2 x 6(1659.24)(12)(10)2x x 6 0(x 199,108.80 199,109Interest charges:323,179.20 150,000 173,179.20 173,179 3)( x 2) 0199,108.80 150,000 49,108.80 49,109x 2 or x 3(c) The vertical asymptote is x 750. The closer thepayment is to 750 per month, the longer the lengthof the mortgage will be. Also, the monthly paymentmust be greater than 750.80. (a) P 3.42 1.297 t ln t2008: P(8) 3.42 1.297(8) ln 8 18.16%2012: P(12) 3.42 1.297(12) ln 12 35.26%50(b)5140(c) Answers will vary. Sample answer: Yes, it can predict the percent for 2020,P( 20) 3.42 1.297( 20) ln 20 74.29% is reasonable. However, for 2030,P(30) 3.42 1.297(30) ln 30 128.92% is not possible.ln 2r81. t 234.727.723.1150000.04 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Section 3.282. t (a)Logarithmic Functions and Their Graphs251ln e number of years required to multiply the original investment by K increases with K. However, the larger the value ofK, the fewer the years required to increase the value of the investment by an additional multiple of the original investment.(b)t605040302010 4 10K4812 16 20 2483. f (t ) 80 17 log(t 1), 0 t 12(a)86.yh10010860012(b) f (0) 80 17 log 1 80.0(c) f ( 4) 80 17 log 5 68.1(d) f (10) 80 17 log 11 62.3 I 84. β 10 log 12 10 4f 10 8(c) No, the difference is due to the logarithmicrelationship between intensity and number ofdecibels.85. False. Reflecting g( x) about the line y x will 4 2x2 2True. The graph of f ( x) ln( x) is a reflection ofh( x) e x in the line y x.87. (a) f ( x) ln x, g ( x) xThe natural log function grows at a slower rate thanthe square root function.40 1 (a) β 10 log 12 10 log(1012 ) 120 decibels 10 10 2 (b) β 10 log 12 10 log(1010 ) 100 decibels 10 6gf010000(b) f ( x) ln x, g ( x) 4xThe natural log function grows at a slower rate thanthe fourth root function.15determine the graph of f ( x).gf0020,000 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
252Chapter 3Exponential and Logarithmic Functions88. (a) The function f ( x) 3x matches graph m, and89.g ( x) log 3 x matches graph n.(b) Since f and g are inverse functions, the point ( a, b)x128y013y is not an exponential function of x, but it is alogarithmic function of x, y log 2 x.on the graph of f corresponds to the point (b, a) onthe graph of g. So, if f ( a) b, then g (b) a.90.x125y2432y is not a logarithmic function of x, but it is anexponential function of x, y 2 x.91. f ( x) (a)ln xxxf ( x)151010 210 410 600.3220.2300.0460.000920.0000138(b) As x , f ( x) 0.0.5(c)0010092. y loga x a y x, so, for example, if a 2, there is no value of y for which ( 2) 4. If a 1, then everyypower of a is equal to 1, so x could only be 1. So, log a x is defined only for 0 a 1 and a 1.Section 3.3 Properties of Logarithms1. change-of-base2.log xln x log aln a3.1log b a4. (a) ln(uv) ln u ln vThis is the Product Property5. (a) log 5 16 (b) log 5 16 6. (a) log1 5 4 (b) log1 5 4 (b) loga u n n loga uThis is the Power Property. u (c) ln ln u ln v v This is the Quotient Property.7. (a) log x(b) log xlog 16log 5ln 16ln 5log 4log(1 5)ln 4ln(1 5)log(3 10)3 10log xln(3 10)3 10ln x8. (a) log 2.6 x (b) log 2.6 x log xlog 2.6ln xln 2.6 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Section 3.3 log 3 7 log 3 52 log 3 7 2 log 3 5log 12ln 12 2.712log 0.4ln 0.410. log 0.4 12 17. log 3log 0.5ln 0.511. logπ 0.5 0.606log πln π( 215) log 1 log 3 7 log 3 518. log 3( 4945 ) log()15. log 3( 257 ) log log 3 5 log 3 32 log 3 7 2 log 3 5 2 log 3 3 2 log 3 7 log 3 5 2 2 log 3 77 log 3 2519. log 3 9 2 log 3 3 2 log 3 7 log 3 52 log 3 7 2 log 3 521. log63 122. log 246 log6148 ( 16 )13log 2 23 3413log620. log5( 16 ) 13 loglog 2 2 34(1) 66 1 3423. log2 ( 2) is undefined. 2 is not in the domain of13( 1)112527. ln e2 ln e5 2 5 728. 2 ln e6 ln e5 ln e12 ln e5 ln24. log3 ( 27) is undefined. 27 is not in the domain of26. lne12e5 ln e7log 3 x .25. ln log5 5 3 3 log 5 5 3(1) 3 13log 2 x .445 log 3 49 log 3 5 log 3 9 log 3 49 log 3 5 log 3 733 log 3 (5 9) log 3 49 log 3 5 log 3 75721 log 3 5 log 3 3 log 3 7 log 3 513. log 3 35 log 3 (5 7)14. log 33 log 3 (3 7) log 3 5log 0.125ln 0.125 5.129log( 2 3)ln( 2 3)12. log 2 3 0.125 25316. log 3 175 log 3 (7 25)log 17ln 17 2.579log 3ln 39. log 3 17 Properties of Logarithms 73e ln e34 34ln e 34(1) 341 ln 1 ln ee1 0 ln e21 0 (1)21 229. log 5 75 log 5 3 log 5753 log 5 25 log 5 52 2 log 5 5 230. log 4 2 log 4 32 log 4 41 2 log 4 45 231. log 4 8 log 4 ( 4 2) log 4 4 log 4 2 log 4 4 log 4 41 2 1 32. log8 16 log8 (8 2) log8 8 log8 2 log8 8 log8 81 3 1 12 12log 4 4 (1) 5252log 4 4(1) 31213 3243 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
254Chapter 3Exponential and Logarithmic Functions33. log b 10 log b 2.540. log b log b 2 log b 53b log b (3b)133 0.3562 0.8271 1.183334. logb23 log b 2 logb 3 0.3562 0.56464100 log b125 log b 1 log b 25 log b 1 log b 52 0 2 log b 5 2(0.8271)12log b 2 12(0.3562) 13(logb 3 13(0.5646 1) 13(1.5646)log b b)42. log 3 13 z log 3 13 log 3 z43. log8 x4 4 log8 x44. ln ( xy ) 3 ln ( xy )3 3(ln x ln y ) 3 ln x 3 ln y2 log b 21 2 log b (3b)41. ln 7 x ln 7 ln x 1.654236. log b13 0.5215 0.208435. log b 0.04 log b 45. log 5 0.178137. log b 45 log b 9.546. log 6 log b 9 log b 55 log 5 5 log 5 xx 1 log 5 x( ) logw2v6w2 log 6 v 2 log 6 w log 6 v2 log b 3 log b 5 2log b 3 log b 5 2(0.5646) 0.8271 1.95632238. logb 3b logb 3 logb b48. ln3t ln t1 3 0.5646 2 2()50. log 4 11b 2c log 4 11 log 4 b 2 log 4 c log 4 11 2 log 4 b log 4 c51. ln z ( z 1) ln z ln( z 1)2 2 log b 2b 2(log b 2 log b b) 2(0.3562 1) 2.7124ln t ln x ln y 2 ln z 2.564639. log b ( 2b)13ln z49. ln xyz 2 ln x ln y ln z 2 logb 3 2 logb b logb 3 2(1)12z ln z1 2 47. ln2 ln z 2 ln( z 1), z 1 x2 1 2352. ln ln ( x 1) ln x3 x ln ( x 1)( x 1) ln x 3 ln ( x 1) ln ( x 1) 3 ln x 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Section 3.3 53. log 2 a2 4 log 2 7 12 log 2 7 12log 2 ( a 2 4) log 2 7 12log 2 ( a 2)( a 2) log 2 7 1 log22 12(a 2) log 2 ( a 2) log 2 7log 2 ( a 2) ln 3 lnx 1 312log 2 ( a 2) log 2 7x2 1259. ln4x 3 ( x 2 3) ln 3 ln ( x 1)2 ln 3 1212ln ( x 2 1) x2 55. log 5 2 3 log 5 x 2 log 5 y 2 z 3 y z 60. lnln x 3 ( x 2 3)1 ln4 1 3 ln4 3457. ln3yz yz ln 2 x2 x 58. log 2 x 41 yz ln 3 x2 1 ln ( yz ) ln x 2 3 1 ln ( yz ) 2 ln x 3 1[ln y ln z 2 ln x]3112ln y ln z ln x333yy log 2 x 4 log 23zz3y1 log 2 x 4 log 2 3z21 log 2 x 4 log 2 y log 2 z 3 213 4 log 2 x log 2 y log 2 z22x ln ( x 2 3) ln x 14ln ( x 2 3)12 ln x( x 2) ln x ln ( x 2)12 ln x 12ln ( x 2)61. ln 3 ln x ln(3x) log x 4 log y 5 log z13x 3 ln ( x 2 3) 12 2 log 5 x 2 log 5 y 3 log 5 zxy 4 log xy 4 log z 5z5 log x log y 4 log z 514x 2 ( x 2) ln x 2 ( x 2) log 5 x 2 (log 5 y 2 log 5 z 3 )56. log255a 2 4 log 2 7 log 2 ( a 2 4) 54. ln Properties of Logarithms62. log 5 8 log5 t log563.238tlog 7 ( z 2) log 7 ( z 2)64. 4 ln 3x ln (3x) ln ln23 41(3 x )4181x 465. log 3 5 x 4 log 3 x log 3 5 x log 3 x 4 5x log 3 4 x 5 log 3 3 x 66. 2 log2 x 4 log2 y log2 x2 log2 y4 log2 x2 y 467. log x 2 log( x 1) log x log( x 1)22 log x( x 1) 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
256Chapter 3Exponential and Logarithmic Functions68. 2 ln 8 5 ln ( z 4) ln 82 ln ( z 4)5 ln 64 ln ( z 4)69. log x 2 log y 3 log z log x log y 2 log z 35 64 ln 5 ( z 4) 70. 3 log 3 x 14 logx log z 3y2 logxz 3y2log 3 y 4 log 3 z log 3 x 3 log 3 y1 4 log 3 z 4() log 3 x 3 4 y log 3 z 4 log 3 y z x3 44x71. ln x ln( x 1) ln( x 1) ln x ln( x 1)( x 1) ln( x 1)( x 1)72. 4 ln z ln ( z 5) 2 ln ( z 5) 4 ln z ( z 5) ln ( z 5)24 ln z ( z 5) ln ( z 5) ln73.z 4 ( z 5)(z 5)2421 122 ln ( x 3) ln x ln ( x 2 1) ln ( x 3) ln x ln ( x 2 1) 2 2 1 2 ln x( x 3) ln ( x 2 1) 2 21 x( x 3) ln 2 2 x 1 21 x( x 3) ln 22 x 1 x( x 3) ln2x2 174. 2 3 ln x ln ( x 1) ln ( x 1) 2 ln x3 ln ( x 1) ln ( x 1) 2 ln x3 ln ( x 1) ln ( x 1) 2 ln x3 ln ( x 1)( x 1) 2 lnx3x 12 x3 ln 2 x 1 75.2112 log8 y 2 log8 ( y 4) log8 ( y 1) log8 y log8 ( y 4) log8 ( y 1) 3 3 12 log8 y( y 4) log8 ( y 1)3 log8 log8 33y ( y 4) log8 ( y 1)22y( y 4) y 1 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Section 3.376.1 log42 (xProperties of Logarithms 1) 2 log 4 ( x 1) 6 log 4 x 1 log42 (x2 1) log 4 ( x 1) log 4 x 6 1 log42 (x2 1)( x 1) log 4 x 6 257 log 4 x 1( x 1) log 4 x 6 log 4 x 6 ( x 1)77. log 2x 1 32log 2 32 log 2 32 log 2 4 4log 2 4The second and third expressions are equal by Property 2.78. log 770 12 12[1 12 12 log 7log 7 70 12[log 7 7 log 7 10]log 7 10]12log 7 1010 by Property 1 and Property 3 I 79. β 10 log 12 10 log I log 10 12 10[log I 12] 120 10 log I 10 When I 10 6 :β 120 10 log 10 6 120 10( 6) 60 decibels I 80. β 10 log 12 10 I 81. β 10 log 12 10 Difference 1.26 10 7 3.16 10 10 10 log 10 log
2018 Cengage Learning. All Rights Reserved. May not be scanned, c opied or duplicated, or poste
