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SECTION 3.1 Exponential FunctionsChapter 3Exponential and Logarithmic FunctionsSection 3.1:Exponential Functions Definition and Graphs of Exponential Functions The Function f (x) e xDefinition and Graphs of Exponential FunctionsDefinition of an Exponential Function:The Graph of an Exponential Function:MATH 1330 Precalculus249
CHAPTER 3 Exponential and Logarithmic Functions250University of Houston Department of Mathematics
SECTION 3.1 Exponential FunctionsExample:Solution:Example:Solution:MATH 1330 Precalculus251
CHAPTER 3 Exponential and Logarithmic FunctionsExample:Solution:Part (a):252University of Houston Department of Mathematics
SECTION 3.1 Exponential FunctionsPart (b):MATH 1330 Precalculus253
CHAPTER 3 Exponential and Logarithmic FunctionsPart (c):Example:Solution:254University of Houston Department of Mathematics
SECTION 3.1 Exponential FunctionsMATH 1330 Precalculus255
CHAPTER 3 Exponential and Logarithmic FunctionsAdditional Example 1:Solution:256University of Houston Department of Mathematics
SECTION 3.1 Exponential FunctionsAdditional Example 2:MATH 1330 Precalculus257
CHAPTER 3 Exponential and Logarithmic FunctionsSolution:258University of Houston Department of Mathematics
SECTION 3.1 Exponential FunctionsAdditional Example 3:Solution:MATH 1330 Precalculus259
CHAPTER 3 Exponential and Logarithmic Functions260University of Houston Department of Mathematics
SECTION 3.1 Exponential FunctionsAdditional Example 4:Solution:MATH 1330 Precalculus261
CHAPTER 3 Exponential and Logarithmic Functions262University of Houston Department of Mathematics
SECTION 3.1 Exponential FunctionsAdditional Example 5:MATH 1330 Precalculus263
CHAPTER 3 Exponential and Logarithmic FunctionsSolution:264University of Houston Department of Mathematics
SECTION 3.1 Exponential FunctionsThe Function f (x) e xExample:Solution:MATH 1330 Precalculus265
CHAPTER 3 Exponential and Logarithmic FunctionsAdditional Example 1:Solution:266University of Houston Department of Mathematics
SECTION 3.1 Exponential FunctionsAdditional Example 2:Solution:MATH 1330 Precalculus267
CHAPTER 3 Exponential and Logarithmic FunctionsAdditional Example 3:Solution:268University of Houston Department of Mathematics
SECTION 3.1 Exponential FunctionsAdditional Example 4:MATH 1330 Precalculus269
CHAPTER 3 Exponential and Logarithmic FunctionsSolution:270University of Houston Department of Mathematics
SECTION 3.1 Exponential FunctionsMATH 1330 Precalculus271
Exercise Set 3.1: Exponential FunctionsSolve for x. (If no solution exists, state ‘No solution.’)1.(a) 3x 2 0(b) 32 x 7 12.(a) 7 x 3 1(b) 7 x 03.(a) 53 x 1 125(b) 253 x 1 1254.(a) 25 x 2 32(b) 85 x 2 325.(a) 24 x 1 8 0(b) 2 x 3 8 06.(a) 7 x 3 1 07(b) 7 2 x 3 5214. Based on your answers to 9-12 above:(a) Draw a sketch of f ( x) 2.7 x withoutplotting points.(b) Draw a sketch of f ( x) 0.73x withoutplotting points.(c) What point on the graph do parts (a) and (b)have in common?(d) Name any asymptote(s) for parts (a) and (b).Answer the following.Graph each of the followingpairs of functions on the same set of axes.15. (a) Graph the functions f ( x) 6 x andx7.(a) 27 x 5 3(b) 36 3 x 0.5 8.(a) 252 x 1 3 5(b)167 7 2 x 149 1 g ( x) on the same set of axes. 6 (b) Compare the graphs drawn in part (a). Whatis the relationship between the graphs?(c) Explain the result from part (b)algebraically.xGraph each of the following functions by plottingpoints. Show any asymptote(s) clearly on your graph.9.f ( x ) 3x10. f ( x) 5x 1 11. f ( x) 2 x 4 12. f ( x) 3 x 5 16. (a) Graph the functions f ( x) and 2 x 2 g ( x) on the same set of axes. 5 (b) Compare the graphs drawn in part (a). Whatis the relationship between the graphs?(c) Explain the result from part (b)algebraically.Sketch a graph of each of the following functions.(Note: You do not need to plot points.) Be sure to labelat least one key point on your graph, and show anyasymptotes.17. f ( x) 8xAnswer the following.13. Based on your answers to 9-12 above:(a) Draw a sketch of f ( x) a x , where a 1 .(b) Draw a sketch of f ( x) a , where0 a 1.(c) What point on the graph do parts (a) and (b)have in common?(d) Name any asymptote(s) for parts (a) and (b).x18. f ( x) 12 x19. f ( x) 0.2 x20. f ( x) 1.4 x 7 21. f ( x) 2 3 22. f ( x) 8 272xxUniversity of Houston Department of Mathematics
Exercise Set 3.1: Exponential FunctionsFor each of the following examples,(a) Find any intercept(s) of the function.(b) Use transformations (the concepts ofreflecting, shifting, stretching, and shrinking)to sketch the graph of the function. Be sure tolabel the transformation of the point 0, 1 .Clearly label any intercepts and/orasymptotes.(c) State the domain and range of the function.(d) State whether the graph is increasing ordecreasing.39. f x 8 1 x 3240. f x 51 x 141. f ( x) 6 x 3642. f ( x) 2 x 1643. f ( x) 2 3x44. f ( x) 12 5x23. f ( x) 2 x 124. f ( x) 3x 2Find the exponential function of the formf ( x ) C a x which satisfies the given conditions.25. f ( x) 5x45. Passes through the points 0, 4 and 3, 32 . 3 26. f ( x) 4 x46. Passes through the points 0, 3 and 2, 108 . 27. f ( x) 8 x47. Passes through the points 0, 2 and 1, 72 .28. f ( x) 10 x7 .48. Passes through the points 0, 7 and 2, 16 x 1 29. f ( x) 4 2 30. f ( x) 4 x 8True or False? (Answer these without using acalculator.)49. e 2 931. f ( x) 9 x 2 2750. e 2 432. f ( x) 7 x 3 751.e 233. f x 4x 1 252.e 134. f x 8x 2 3253. e 1 054. e 1 035. f ( x) 9 3x 436. f ( x) 25 5x 32 x37. f ( x) 338. f ( x) 4 3 x55. e3 2756. e 5257. e 0 158. e1 0MATH 1330 Precalculus273
Exercise Set 3.1: Exponential FunctionsAnswer the following.59. Graph f ( x) 2 x and g ( x) e x on the same setof axes.71. f ( x) e x 4 372. f ( x) e x 3 260. Graph f ( x) e x and g ( x) 3x on the same setof axes.61. Sketch the graph of f ( x) e x and then find thefollowing:(a)(b)(c)(d)DomainRangeHorizontal Asymptotey-intercept62. Sketch the graph of f ( x) 3x and then find thefollowing:(a)(b)(c)(d)(e)DomainRangeHorizontal Asymptotey-interceptHow do your answers compare with those inthe previous question?Graph the following functions, not by plotting points,but by applying transformations to the graph ofy e x . Be sure to label at least one key point on yourgraph (the transformation of the point (0, 1)) and showany asymptotes. Then state the domain and range ofthe function.63. f ( x) 3e x64. f ( x) 2e x65. f ( x) e x 1 666. f ( x) e x 2 167. f ( x) e x 368. f ( x) e x 269. f ( x) 4e x 570. f ( x) 2e x 6 4274University of Houston Department of Mathematics
SECTION 3.2 Logarithmic FunctionsSection 3.2:Logarithmic Functions Evaluating Logarithms Graphing Logarithmic FunctionsEvaluating LogarithmsExample:Solution:MATH 1330 Precalculus275
CHAPTER 3 Exponential and Logarithmic ersity of Houston Department of Mathematics
SECTION 3.2 Logarithmic FunctionsExample:Solution:MATH 1330 Precalculus277
CHAPTER 3 Exponential and Logarithmic FunctionsAdditional Example 1:Solution:278University of Houston Department of Mathematics
SECTION 3.2 Logarithmic FunctionsMATH 1330 Precalculus279
CHAPTER 3 Exponential and Logarithmic FunctionsAdditional Example 2:Solution:280University of Houston Department of Mathematics
SECTION 3.2 Logarithmic FunctionsAdditional Example 3:Solution:MATH 1330 Precalculus281
CHAPTER 3 Exponential and Logarithmic Functions282University of Houston Department of Mathematics
SECTION 3.2 Logarithmic FunctionsAdditional Example 4:Solution:MATH 1330 Precalculus283
CHAPTER 3 Exponential and Logarithmic FunctionsAdditional Example 5:Solution:284University of Houston Department of Mathematics
SECTION 3.2 Logarithmic FunctionsGraphing Logarithmic FunctionsThe Graph of a Logarithmic Function:MATH 1330 Precalculus285
CHAPTER 3 Exponential and Logarithmic FunctionsLogarithmic and Exponential Functions as Inverses:286University of Houston Department of Mathematics
SECTION 3.2 Logarithmic FunctionsExample:Solution:Example:Solution:MATH 1330 Precalculus287
CHAPTER 3 Exponential and Logarithmic FunctionsExample:Solution:Part (a):288University of Houston Department of Mathematics
SECTION 3.2 Logarithmic FunctionsPart (b):Part (c):Additional Example 1:Solution:MATH 1330 Precalculus289
CHAPTER 3 Exponential and Logarithmic FunctionsAdditional Example 2:290University of Houston Department of Mathematics
SECTION 3.2 Logarithmic FunctionsSolution:MATH 1330 Precalculus291
CHAPTER 3 Exponential and Logarithmic FunctionsAdditional Example 3:Solution:292University of Houston Department of Mathematics
SECTION 3.2 Logarithmic FunctionsMATH 1330 Precalculus293
CHAPTER 3 Exponential and Logarithmic FunctionsAdditional Example 4:Solution:294University of Houston Department of Mathematics
SECTION 3.2 Logarithmic FunctionsAdditional Example 5:Solution:MATH 1330 Precalculus295
CHAPTER 3 Exponential and Logarithmic Functions296University of Houston Department of Mathematics
Exercise Set 3.2: Logarithmic FunctionsWrite each of the following equations in exponentialform.(b) ln e 2 (b) ln26. (a) ln e 4(b) ln 1 27. (a) ln (b) ln e x28. (a) ln (b) ln e2 x(a) log3 81 4(b) log 4 8 2.(a) log5 1 0(b) log 3 19 225. (a) ln e 63.(a) log5 7 x(b) ln x 84.(a) log x 2 3(b) ln x 2Write each of the following equations in logarithmicform.1(b) 10 4 10,0005.(a) 60 16.(a) 5 125(b) 257.(a) 7 1 7(b) e 4 x8.(a) 100.5 10(b) e 5 y9.3(a) e 7x 3(b) e10. (a) e x 121 1252x 2 y(b) ln e4 1.32 23. (a) ln e 24. (a) ln e5 e 3 1e4 1eSimplify each of the following expressions.29. (a) 7log7 5 (b) 10log 3 5 log 430. (a) 10 (b) 5ln 631. (a) e (b) eln 232. (a) e ln x(b) e log0.71 ln x 4(b) e5 x y 6Simplify each of the following expressions.Find the value of x in each of the following equations.Write all answers in simplest form. (Some answersmay contain e.)11. (a) log 2 8 (b) log5 1 33. (a) log 2 32 x(b) log x 9 212. (a) log3 81 (b) log8 8 34. (a) log5 125 x(b) log x 64 313. (a) log 7 7 (b) log 6 6235. (a) log 7 x 2(b) log x 3 14. (a) log9 1 (b) log 2 2515. (a) log16 4 (b) log 7 17 36. (a) log3 x 4(b) log x 18 137. (a) log 25 125 x(b) log16 x 38. (a) log16 32 x(b) log36 x 0.539. (a) log x 3 2(b) log3 log 4 x 016. (a) log 110 (b) log32 2 17. (a) log 27 3 1(b) log 9 81 18. (a) log 49 7 (b) log 2 18 19. (a) log 4 0.25 (b) log 10 4 7 (b) log5 0.2 21. (a) log 4 0.5 (b) log8 32 20. (a) log 722. (a) log 256 5 MATH 1330 Precalculus(b) log 32 116 1214 40. (a) log x 2 9 3 (b) log 5 log 2 x 141. (a) ln x 4(b) ln x 5 242. (a) ln x 0(b) ln x 2 5297
Exercise Set 3.2: Logarithmic FunctionsSolve for x. Give an exact answer, and then use acalculator to round that answer to the nearestthousandth. (Hint: Write the equations inlogarithmic form first.)For each of the following examples,(a) Use transformations (the concepts ofreflecting, shifting, stretching, and shrinking)to sketch the graph of y logb x . Be sure to43. (a) 10 x 20(b) e x 20label the transformation of the point 1, 0 .44. (a) 10 x 45(b) e x 4545. (a) 10 x 25(b) e x 25Clearly label any asymptotes.(b) State the domain and range of the function.(c) State whether the graph is increasing ordecreasing.46. (a) 10 x 47(b) e x 4755. f ( x) log 2 x 347. 103 x 2 656. f ( x) log5 x 448. e5 x 1 757. f ( x) log3 x 49. e4 x250. 10 x 402 1 758. f ( x ) log 6 ( x)59. f ( x) log 7 ( x 5)60. f ( x) log8 ( x 2)Answer the following.51. (a) Sketch the graph of f ( x) 2 x .(b) Sketch the graph of f 1 ( x) on the same setof axes.(c) The inverse of y 2 x is y .52. (a) Sketch the graph of f ( x) e x .(b) Sketch the graph of f 1 ( x) on the same setof axes.(c) The inverse of y e x is y .61. f ( x) log 5 ( x 1) 462. f ( x) log 6 ( x 4) 263. f ( x) ln( x )64. f ( x) ln x 65. f ( x) ln( x 4)66. f ( x) ln( x) 167. f ( x) 5 ln( x)68. f ( x) 3 ln( x 2)53. (a) Sketch the graph of y logb x , whereb 1.(b) Name any intercept(s) for the graph ofy logb x .(c) Name any asymptote(s) for the graph ofy logb x .54. (a) Sketch the graph of y ln x .(b) y ln x can be written as y logb x ,where b .(c) Name any intercept(s) for the graph ofy ln x .(d) Name any asymptote(s) for the graph ofy ln x .298Find the domain of each of the following functions.(Note: You should be able to do this algebraically,without sketching the graph.)69. f ( x) log8 x 70. f ( x) log5 x 71. f ( x) 2ln x 372. f ( x) 3ln(2 x)73. f ( x) log 6 ( x 2)74. f ( x) log10 ( x 3)75. f ( x) log10 ( x) 3University of Houston Department of Mathematics
Exercise Set 3.2: Logarithmic Functions76. f ( x) log 3 (3 4 x) 577. f ( x) ln(5 2 x) 178. f ( x) ln( 4 x) 8 79. f ( x) log 4 x 2 180. f ( x) log 7 x 2 9 81. f ( x) log 2 x 2 x 6 82. f ( x) log9 x 2 4 MATH 1330 Precalculus299
CHAPTER 3 Exponential and Logarithmic FunctionsSection 3.3:Laws of Logarithms Laws of LogarithmsLaws of LogarithmsThree Laws of Logarithms:Verifying Laws of Logarithms:300University of Houston Department of Mathematics
SECTION 3.3 Laws of LogarithmsExa
MATH 1330 Precalculus 273 For each of the following examples, (a) Find any intercept(s) of the function. (b) Use transformations (the concepts of reflecting, shifting, stretching, and shrinking) to sketch the graph of the function. Be sure to label the transformation of the point 0,1 ( ) 6 36 . Clearly label any intercepts and/or asymptotes. (c) State the domain and range of the function. (d .
