Exponential And Logarithmic Functions 3
Exponential andLogarithmic Functions3.1Exponential Functions and Their Graphs3.2Logarithmic Functions and Their Graphs3.3Properties of Logarithms3.4Exponential and Logarithmic Equations3.5Exponential and Logarithmic Models3 Sylvain Grandadam/Getty ImagesCarbon dating is a method used todetermine the ages of archeologicalartifacts up to 50,000 years old. Forexample, archeologists are usingcarbon dating to determine the agesof the great pyramids of Egypt.S E L E C T E D A P P L I C AT I O N SExponential and logarithmic functions have many real-life applications. The applications listed belowrepresent a small sample of the applications in this chapter. Computer Virus,Exercise 65, page 227 Galloping Speeds of Animals,Exercise 85, page 244 IQ Scores,Exercise 47, page 266 Data Analysis: Meteorology,Exercise 70, page 228 Average Heights,Exercise 115, page 255 Forensics,Exercise 63, page 268 Sound Intensity,Exercise 90, page 238 Carbon Dating,Exercise 41, page 266 Compound Interest,Exercise 135, page 273217
218Chapter 33.1Exponential and Logarithmic FunctionsExponential Functions and Their GraphsWhat you should learn Recognize and evaluate exponential functions with base a. Graph exponential functionsand use the One-to-OneProperty. Recognize, evaluate, and graphexponential functions withbase e. Use exponential functions tomodel and solve real-lifeproblems.Why you should learn itExponential functions can beused to model and solve real-lifeproblems. For instance, inExercise 70 on page 228, anexponential function is used tomodel the atmospheric pressureat different altitudes.Exponential FunctionsSo far, this text has dealt mainly with algebraic functions, which include polynomial functions and rational functions. In this chapter, you will study two typesof nonalgebraic functions—exponential functions and logarithmic functions.These functions are examples of transcendental functions.Definition of Exponential FunctionThe exponential function f with base a is denoted byf sxd 5 a xwhere a 0, a Þ 1, and x is any real number.The base a 5 1 is excluded because it yields f sxd 5 1x 5 1. This is a constantfunction, not an exponential function.You have evaluated a x for integer and rational values of x. For example, youknow that 43 5 64 and 41y2 5 2. However, to evaluate 4x for any real number x,you need to interpret forms with irrational exponents. For the purposes of thistext, it is sufficient to think ofa!2(where !2 1.41421356)as the number that has the successively closer approximationsa1.4, a1.41, a1.414, a1.4142, a1.41421, . . . .Example 1Evaluating Exponential FunctionsUse a calculator to evaluate each function at the indicated value of x.Functiona. f sxd 5 2 xb. f sxd 5 22xc. f sxd 5 0.6xSolution Graphing Calculator Keystrokesx2 c 3.1 ENTER2x2 c p ENTER2x 3 4 2 dENTER.6 Function Valuea. f s23.1d 5 223.1b. f spd 5 22pc. f s32 d 5 s0.6d3y2 Comstock Images/AlamyValuex 5 23.1x5px 5 23Display0.11662910.11331470.4647580Now try Exercise 1.The HM mathSpace CD-ROM andEduspace for this text containadditional resources related to theconcepts discussed in this chapter.When evaluating exponential functions with a calculator, remember toenclose fractional exponents in parentheses. Because the calculator follows theorder of operations, parentheses are crucial in order to obtain the correct result.
Section 3.1219Graphs of Exponential FunctionsExplorationNote that an exponential functionf sxd 5 a x is a constant raisedto a variable power, whereas apower function gsxd 5 x n is avariable raised to a constantpower. Use a graphing utilityto graph each pair of functionsin the same viewing window.Describe any similarities anddifferences in the graphs.a. y1 5 2x, y2 5 x2b. y1 5 3x, y2 5 x3yExponential Functions and Their GraphsThe graphs of all exponential functions have similar characteristics, as shown inExamples 2, 3, and 5.Example 2Graphs of y 5 axIn the same coordinate plane, sketch the graph of each function.a. f sxd 5 2xb. gsxd 5 4xSolutionThe table below lists some values for each function, and Figure 3.1 shows thegraphs of the two functions. Note that both graphs are increasing. Moreover, thegraph of gsxd 5 4x is increasing more rapidly than the graph of f sxd 5 2x.g(x) 4xx2322210122x1816414121412414164x1611614Now try Exercise 11.1210The table in Example 2 was evaluated by hand. You could, of course, use agraphing utility to construct tables with even more values.864f(x) 2x2Example 3Graphs of y 5 a –xx 4 3 2 1 2FIGURE1234In the same coordinate plane, sketch the graph of each function.a. F sxd 5 22x3.1b. G sxd 5 42xSolutionThe table below lists some values for each function, and Figure 3.2 shows thegraphs of the two functions. Note that both graphs are decreasing. Moreover, thegraph of G sxd 5 42x is decreasing more rapidly than the graph of F sxd 5 22x.yG(x) 4 x) 2 xx 4 3 2 1 2FIGURE3.21234Now try Exercise 13.In Example 3, note that by using one of the properties of exponents, the functions F sxd 5 22x and Gsxd 5 42x can be rewritten with positive exponents.F sxd 5 22x 5121152x2xand Gsxd 5 42x 51215 14x 4x
220Chapter 3Exponential and Logarithmic FunctionsComparing the functions in Examples 2 and 3, observe thatFsxd 5 22x 5 f s2xdGsxd 5 42x 5 gs2xd.andConsequently, the graph of F is a reflection (in the y-axis) of the graph of f. Thegraphs of G and g have the same relationship. The graphs in Figures 3.1 and 3.2are typical of the exponential functions y 5 a x and y 5 a2x. They have oney-intercept and one horizontal asymptote (the x-axis), and they are continuous.The basic characteristics of these exponential functions are summarized inFigures 3.3 and 3.4.yNotice that the range of anexponential function is s0, d,which means that a x 0 for allvalues of x.y ax(0, 1)xFIGURE3.3yy a x(0, 1)xFIGUREGraph of y 5 a x, a 1 Domain: s2 , d Range: s0, d Intercept: s0, 1d Increasing x-axis is a horizontal asymptotesax 0 as x 2 d ContinuousGraph of y 5 a2x, a 1 Domain: s2 , d Range: s0, d Intercept: s0, 1d Decreasing x-axis is a horizontal asymptotesa2x 0 as x d Continuous3.4From Figures 3.3 and 3.4, you can see that the graph of an exponentialfunction is always increasing or always decreasing. As a result, the graphs passthe Horizontal Line Test, and therefore the functions are one-to-one functions.You can use the following One-to-One Property to solve simple exponentialequations.For a 0 and a Þ 1, ax 5 ay if and only if x 5 y.Example 4Using the One-to-One Propertya. 9 5 3x1132 5 3x1125x1115xb.1 x2sdOne-to-One PropertyOriginal equation9 5 32One-to-One PropertySolve for x.5 8 22x 5 23 x 5 23Now try Exercise 45.
Section 3.1221Exponential Functions and Their GraphsIn the following example, notice how the graph of y 5 a x can be used tosketch the graphs of functions of the form f sxd 5 b a x1c.Transformations of Graphs of Exponential FunctionsExample 5Each of the following graphs is a transformation of the graph of f sxd 5 3x.a. Because gsxd 5 3x11 5 f sx 1 1d, the graph of g can be obtained by shiftingthe graph of f one unit to the left, as shown in Figure 3.5.b. Because hsxd 5 3x 2 2 5 f sxd 2 2, the graph of h can be obtained byshifting the graph of f downward two units, as shown in Figure 3.6.c. Because ksxd 5 23x 5 2f sxd, the graph of k can be obtained by reflectingthe graph of f in the x-axis, as shown in Figure 3.7.d. Because j sxd 5 32x 5 f s2xd, the graph of j can be obtained by reflecting thegraph of f in the y-axis, as shown in Figure 3.8.yy23f (x) 3 xg(x) 3 x 112x 21 11f(x) 3 x2 1h(x) 3 x 2x 2FIGURE3.5 21 1Horizontal shiftFIGURE3.6Vertical shiftyy4213f(x) 3 xx1 2 12k(x) 3 x2j(x) 3 xf(x) 3 x1x 2 2FIGURE3.7Reflection in x-axisFIGURE 13.812Reflection in y-axisNow try Exercise 17.Notice that the transformations in Figures 3.5, 3.7, and 3.8 keep the x-axisas a horizontal asymptote, but the transformation in Figure 3.6 yields a newhorizontal asymptote of y 5 22. Also, be sure to note how the y-intercept isaffected by each transformation.
222Chapter 3Exponential and Logarithmic Functionsy3The Natural Base eIn many applications, the most convenient choice for a base is the irrationalnumber(1, e)e 2.718281828 . . . .2f(x) ( 1, e 1)This number is called the natural base. The function given by f sxd 5 e x is calledthe natural exponential function. Its graph is shown in Figure 3.9. Be sureyou see that for the exponential function f sxd 5 e x, e is the constant2.718281828 . . . , whereas x is the variable.ex(0, 1)( 2, e 2)x 2FIGURE 11Exploration3.9Use a graphing utility to graph y1 5 s1 1 1yxd x and y2 5 e in the sameviewing window. Using the trace feature, explain what happens to the graphof y1 as x increases.Example 6Use a calculator to evaluate the function given by f sxd 5 e x at each indicatedvalue of x.a. x 5 22b. x 5 21c. x 5 0.25d. x 5 20.3y8f(x) 2e 0.24x7Evaluating the Natural Exponential FunctionSolution654a.b.c.d.31Function Valuef s22d 5 e22f s21d 5 e21f s0.25d 5 e0.25f s20.3d 5 e20.3Graphing Calculator Keystrokesex x2 c 2 ENTERex x2 c 1 ENTERex 0.25 ENTERex x2 c 0.3 ENTERDisplay0.13533530.36787941.28402540.7408182x 4 3 2 1FIGURE123Now try Exercise 27.43.10Example 7ySketch the graph of each natural exponential function.a. f sxd 5 2e0.24x87To sketch these two graphs, you can use a graphing utility to construct a table ofvalues, as shown below. After constructing the table, plot the points and connectthem with smooth curves, as shown in Figures 3.10 and 3.11. Note that the graphin Figure 3.10 is increasing, whereas the graph in Figure 3.11 is decreasing.543g(x) 12 e 0.58xxFIGURE3.112322210123f 950.8930.5000.2800.1570.088x1 4 3 2 1b. gsxd 5 12e20.58xSolution62Graphing Natural Exponential Functions1234Now try Exercise 35.
Section 3.1Use the formula1223ApplicationsExplorationA5P 11Exponential Functions and Their Graphsrn2ntto calculate the amount in anaccount when P 5 3000,r 5 6%, t 5 10 years, andcompounding is done (a) by theday, (b) by the hour, (c) by theminute, and (d) by the second.Does increasing the number ofcompoundings per year result inunlimited growth of the amountin the account? Explain.One of the most familiar examples of exponential growth is that of an investmentearning continuously compounded interest. Using exponential functions, you candevelop a formula for interest compounded n times per year and show how itleads to continuous compounding.Suppose a principal P is invested at an annual interest rate r, compoundedonce a year. If the interest is added to the principal at the end of the year, the newbalance P1 isP1 5 P 1 Pr5 Ps1 1 rd.This pattern of multiplying the previous principal by 1 1 r is then repeated eachsuccessive year, as shown below.Year0123.Balance After Each CompoundingP5PP1 5 Ps1 1 rdP2 5 P1s1 1 rd 5 Ps1 1 rds1 1 rd 5 Ps1 1 rd2P3 5 P2s1 1 rd 5 Ps1 1 rd2s1 1 rd 5 Ps1 1 rd3.Pt 5 Ps1 1 rdttTo accommodate more frequent (quarterly, monthly, or daily) compoundingof interest, let n be the number of compoundings per year and let t be the number of years. Then the rate per compounding is ryn and the account balance aftert years is1A5P 11rn2.ntAmount (balance) with n compoundings per yearIf you let the number of compoundings n increase without bound, the processapproaches what is called continuous compounding. In the formula for ncompoundings per year, let m 5 nyr. This produces1rn5P 111rmr11mA5P 11m11 1 m1 .7182804692.718281693 em5P 11315P112ntAmount with n compoundings per year22mrtSubstitute mr for n.mrt1mSimplify.24.m rtProperty of exponentsAs m increases without bound, the table at the left shows that f1 1 s1ymdgm eas m . From this, you can conclude that the formula for continuouscompounding isA 5 Pert.Substitute e for s1 1 1ymdm.
224Chapter 3Exponential and Logarithmic FunctionsFormulas for Compound InterestBe sure you see that the annualinterest rate must be written indecimal form. For instance, 6%should be written as 0.06.After t years, the balance A in an account with principal P and annualinterest rate r (in decimal form) is given by the following formulas.11. For n compoundings per year: A 5 P 1 1rn2nt2. For continuous compounding: A 5 Pe rtCompound InterestExample 8A total of 12,000 is invested at an annual interest rate of 9%. Find the balanceafter 5 years if it is compoundeda. quarterly.b. monthly.c. continuously.Solutiona. For quarterly compounding, you have n 5 4. So, in 5 years at 9%, thebalance is1A5P 11rn2ntFormula for compound interest15 12,000 1 10.09424(5)Substitute for P, r, n, and t. 18,726.11.Use a calculator.b. For monthly compounding, you have n 5 12. So, in 5 years at 9%, thebalance is1A5P 11rn2nt15 12,000 1 1Formula for compound interest0.0912212(5) 18,788.17.Substitute for P, r, n, and t.Use a calculator.c. For continuous compounding, the balance isA 5 Pe rtFormula for continuous compounding5 12,000e0.09(5)Substitute for P, r, and t. 18,819.75.Use a calculator.Now try Exercise 53.In Example 8, note that continuous compounding yields more than quarterlyor monthly compounding. This is typical of the two types of compounding. Thatis, for a given principal, interest rate, and time, continuous compounding willalways yield a larger balance than compounding n times a year.
Section 3.1Example 9Plutonium (in pounds)P10987654321( 12( t/24,100(24,100, 5)P 5 10(100,000, 0.564)t50,000100,000Years of decayFIGURE3.12Radioactive DecayIn 1986, a nuclear reactor accident occurred in Chernobyl in what was then theSoviet Union. The explosion spread highly toxic radioactive chemicals, such asplutonium, over hundreds of square miles, and the government evacuated the cityand the surrounding area. To see why the city is now uninhabited, consider themodelRadioactive DecayP 10225Exponential Functions and Their Graphs1122ty24,100which represents the amount of plutonium P that remains (from an initial amountof 10 pounds) after t years. Sketch the graph of this function over the intervalfrom t 5 0 to t 5 100,000, where t 5 0 represents 1986. How much of the 10pounds will remain in the year 2010? How much of the 10 pounds will remainafter 100,000 years?SolutionThe graph of this function is shown in Figure 3.12. Note from this graph thatplutonium has a half-life of about 24,100 years. That is, after 24,100 years, halfof the original amount will remain. After another 24,100 years, one-quarter of theoriginal amount will remain, and so on. In the year 2010 st 5 24d, there will stillbe12P 5 10121224y24,100 100.000995912 9.993 poundsof plutonium remaining. After 100,000 years, there will still be11221122100,000y24,1004.1494 10P 5 10 0.564 poundof plutonium remaining.Now try Exercise 67.WRITING ABOUTMATHEMATICSIdentifying Exponential Functions Which of the following functions generated thetwo tables below? Discuss how you were able to decide. What do these functionshave in common? Are any of them the same? If so, explain why.a. f1sxd 5 2(x13)b. f2sxd 5 8s 12 dc. f3sxd 5 s 12 dd. f4sxd 5 s 12 d 1 7e. f5sxd 5 7 1 2xf. f6sxd 5 2Create two different exponential functions of the forms y 5 asbdx and y 5 c x 1 dwith y-intercepts of s0, 23d.
226Chapter 33.1Exponential and Logarithmic FunctionsThe HM mathSpace CD-ROM and Eduspace for this text contain step-by-step solutionsto all odd-numbered exercises. They also provide Tutorial Exercises for additional help.ExercisesVOCABULARY CHECK: Fill in the blanks.1. Polynomials and rational functions are examples of functions.2. Exponential and logarithmic functions are examples of nonalgebraic functions, also called functions.3. The exponential function given by f sxd 5 e x is called the function, and the base eis called the base.4. To find the amount A in an account after t years with principal P and an annual interest rate r compoundedn times per year, you can use the formula .5. To find the amount A in an account after t years with principal P and an annual interest rate r compoundedcontinuously, you can use the formula .PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.In Exercises 1– 6, evaluate the function at the indicatedvalue of x. Round your result to three decimal places.Function1. f sxd 5Value3.4xx 5 233. f sxd 5 5xx 5 2p24. f sxd 5 s3d3x 5 105. g sxd 5 5000s2xdx 5 21.56. f sxd 5 200s1.2d12xx 5 245xy442 24666442 27. f sxd 5 2x9. f sxd 5 22x4 22 28. f sxd 5 2x 1 110. f sxd 5 2x223x2211Functionx 424. y 5 32 x 226. y 5 4x11 2 2In Exercises 27–32, evaluate the function at the indicatedvalue of x. Round your result to three decimal places.x 22x16In Exercises 23–26, use a graphing utility to graph theexponential function.25. y 5y(d)2 419. f sxd 5 22x, gsxd 5 5 2 2 x23. y 5 22x 2y18. f sxd 5 4x, gsxd 5 4x 1 122. f sxd 5 0.3x, gsxd 5 20.3x 1 5x 2(c)16. f sxd 5 4x23 1 37721. f sxd 5 s2d , gsxd 5 2s2dx415. f sxd 5x6214. f sxd 5 6x2 x2120. f sxd 5 10 x, gsxd 5 102 x13y(b)6 213. f sxd 5 62x2x17. f sxd 5 3 x, gsxd 5 3x242 4112. f sxd 5 s2dIn Exercises 17–22, use the graph of f to describe thetransformation that yields the graph of g.In Exercises 7–10, match the exponential function with itsgraph. [The graphs are labeled (a), (b), (c), and (d).](a)111. f sxd 5 s2dxx 5 5.62. f sxd 5 2.3xIn Exercises 11–16, use a graphing utility to construct atable of values for the function. Then sketch the graph ofthe function.427. hsxd 5e2xValuex 5 3428. f sxd 5 e xx 5 3.229. f sxd 5x 5 102e25x30. f sxd 5 1.5e xy231. f sxd 55000e0.06x32. f sxd 5 250e0.05xx 5 240x56x 5 20
Section 3.1In Exercises 33–38, use a graphing utility to construct atable of values for the function. Then sketch the graph ofthe function.33. f sxd 5 e x34. f sxd 5 e 2x35. f sxd 5 3e x1436. f sxd 5 2e20.5x37. f sxd 52e x2238. f sxd 5 2 1 e x2514In Exercises 39– 44, use a graphing utility to graph theexponential function.39. y 5 1.0825x40. y 5 1.085x41. sstd 5 2e0.12t42. sstd 5 3e20.2t43. gsxd 5 1 1 e2x44. hsxd 5 e x22In Exercise 45–52, use the One-to-One Property to solve theequation for x.45. 3x11 5 2747. 2x22 546. 2x23 5 1648.49. e3x12 5 e3251. ex231152x111325 12550. e2x21 5 e45 e2x52. ex2165 e5xCompound Interest In Exercises 53–56, complete thetable to determine the balance A for P dollars invested atrate r for t years and compounded n times per year.n12412365ContinuousAExponential Functions and Their Graphs22762. Trust Fund A deposit of 5000 is made in a trust fundthat pays 7.5% interest, compounded continuously. It isspecified that the balance will be given to the college fromwhich the donor graduated after the money has earnedinterest for 50 years. How much will the college receive?63. Inflation If the annual rate of inflation averages 4% overthe next 10 years, the approximate costs C of goods orservices during any year in that decade will be modeled byCstd 5 Ps1.04d t, where t is the time in years and P is thepresent cost. The price of an oil change for your car ispresently 23.95. Estimate the price 10 years from now.The demand equation for a product is given by64. Demand1p 5 5000 1 244 1 e20.002x2where p is the price and x is the number of units.(a) Use a graphing utility to graph the demand function forx 0 and p 0.(b) Find the price p for a demand of x 5 500 units.(c) Use the graph in part (a) to approximate the greatestprice that will still yield a demand of at least 600 units.65. Computer Virus The number V of computers infected bya computer virus increases according to the modelVstd 5 100e4.6052t, where t is the time in hours. Find (a) Vs1d,(b) Vs1.5d, and (c) Vs2d.66. Population The population P (in millions) of Russiafrom 1996 to 2004 can be approximated by the modelP 5 152.26e20.0039t, where t represents the year, with t 5 6corresponding to 1996. (Source: Census Bureau,International Data Base)53. P 5 2500, r 5 2.5%, t 5 10 years(a) According to the model, is the population of Russiaincreasing or decreasing? Explain.54. P 5 1000, r 5 4%, t 5 10 years(b) Find the population of Russia in 1998 and 2000.55. P 5 2500, r 5 3%, t 5 20 years(c) Use the model to predict the population of Russia in2010.56. P 5 1000, r 5 6%, t 5 40 yearsCompound Interest In Exercises 57– 60, complete thetable to determine the balance A for 12,000 invested atrate r for t years, compounded continuously.t1020304050A67. Radioactive Decay Let Q represent a mass of radioactiveradium s226Rad (in grams), whose half-life is 1599 years.The quantity of radium present after t years isty1599Q 5 25s12 d.(a) Determine the initial quantity (when t 5 0).(b) Determine the quantity present after 1000 years.(c) Use a graphing utility to graph the function over theinterval t 5 0 to t 5 5000.57. r 5 4%58. r 5 6%59. r 5 6.5%60. r 5 3.5%61. Trust Fund On the day of a child’s birth, a deposit of 25,000 is made in a trust fund that pays 8.75% interest,compounded continuously. Determine the balance in thisaccount on the child’s 25th birthday.68. Radioactive Decay Let Q represent a mass of carbon14 s14Cd (in grams), whose half-life is 5715 years. The quan1 ty5715.tity of carbon 14 present after t years is Q 5 10s2 d(a) Determine the initial quantity (when t 5 0).(b) Determine the quantity present after 2000 years.(c) Sketch the graph of this function over the interval t 5 0to t 5 10,000.
228Chapter 3Exponential and Logarithmic FunctionsSynthesisModel It69. Data Analysis: Biology To estimate the amount ofdefoliation caused by the gypsy moth during a givenyear, a forester counts the number x of egg masses on140 of an acre (circle of radius 18.6 feet) in the fall. Thepercent of defoliation y the next spring is shown in thetable. (Source: USDA, Forest Service)Egg masses, xPercent of defoliation, y02550751001244819699True or False? In Exercises 71 and 72, determine whetherthe statement is true or false. Justify your answer.71. The line y 5 22 is an asymptote for the graph off sxd 5 10 x 2 2.72. e 5271,801.99,990Think About It In Exercises 73–76, use properties of exponents to determine which functions (if any) are the same.73. f sxd 5 3x22gsxd 5 3x 2 9hsxd 5gsxd 5 22x16hsxd 5 64s4xd1 x9s3 d75. f sxd 5 16s42xdgsxd 5 sA model for the data is given by74. f sxd 5 4x 1 1276. f sxd 5 e2x 1 3d1 x224gsxd 5 e32xhsxd 5 16s222xd100y5.1 1 7e20.069x(a) Use a graphing utility to create a scatter plot of thedata and graph the model in the same viewingwindow.(b) Create a table that compares the model with thesample data.(c) Estimate the percent of defoliation if 36 egg masses1are counted on 40 acre.(d) You observe that 23 of a forest is defoliated thefollowing spring. Use the graph in part (a) to1estimate the number of egg masses per 40acre.70. Data Analysis: Meteorology A meteorologist measuresthe atmospheric pressure P (in pascals) at altitude h (inkilometers). The data are shown in the table.77. Graph the functions given by y 5 3x and y 5 4x and use thegraphs to solve each inequality.(a) 4x 3xPressure, P05101520101,29354,73523,29412,1575,069A model for the data is given by P 5 107,428e 20.150h.(b) 4x 3x78. Use a graphing utility to graph each function. Use thegraph to find where the function is increasing anddecreasing, and approximate any relative maximum orminimum values.(b) gsxd 5 x232x(a) f sxd 5 x 2e2x79. Graphical Analysis1f sxd 5 1 10.5x2Use a graphing utility to graphxgsxd 5 e0.5andin the same viewing window. What is the relationshipbetween f and g as x increases and decreases withoutbound?80. Think About It Which functions are exponential?(a) 3xAltitude, hhsxd 5 2e x23(b) 3x 2(c) 3x(d) 22xSkills ReviewIn Exercises 81 and 82, solve for y.81. x 2 1 y 2 5 25 82. x 2 y 5 2In Exercises 83 and 84, sketch the graph of the function.83. f sxd 5291x84. f sxd 5 !7 2 x(a) Sketch a scatter plot of the data and graph the model onthe same set of axes.(b) Estimate the atmospheric pressure at a height of8 kilometers.85. Make a Decision To work an extended applicationanalyzing the population per square mile of the UnitedStates, visit this text’s website at college.hmco.com. (DataSource: U.S. Census Bureau)
Section 3.23.2Logarithmic Functions and Their Graphs229Logarithmic Functions and Their GraphsWhat you should learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions. Recognize, evaluate, and graphnatural logarithmic functions. Use logarithmic functions tomodel and solve real-lifeproblems.Logarithmic FunctionsIn Section 1.9, you studied the concept of an inverse function. There, you learnedthat if a function is one-to-one—that is, if the function has the property that nohorizontal line intersects the graph of the function more than once—the functionmust have an inverse function. By looking back at the graphs of the exponentialfunctions introduced in Section 3.1, you will see that every function of the formf sxd 5 a x passes the Horizontal Line Test and therefore must have an inversefunction. This inverse function is called the logarithmic function with base a.Why you should learn itLogarithmic functions are oftenused to model scientific observations. For instance, in Exercise89 on page 238, a logarithmicfunction is used to modelhuman memory.Definition of Logarithmic Function with Base aFor x 0, a 0, and a Þ 1,y 5 loga x if and only if x 5 a y.The function given byf sxd 5 loga xRead as “log base a of x.”is called the logarithmic function with base a.The equationsy 5 loga x Ariel Skelley/Corbisandx 5 ayare equivalent. The first equation is in logarithmic form and the second is inexponential form. For example, the logarithmic equation 2 5 log3 9 can berewritten in exponential form as 9 5 32. The exponential equation 53 5 125 canbe rewritten in logarithmic form as log5 125 5 3.When evaluating logarithms, remember that a logarithm is an exponent.This means that loga x is the exponent to which a must be raised to obtain x. Forinstance, log2 8 5 3 because 2 must be raised to the third power to get 8.Example 1Evaluating LogarithmsUse the definition of logarithmic function to evaluate each logarithm at the indicated value of x.Remember that a logarithm isan exponent. So, to evaluate thelogarithmic expression loga x,you need to ask the question,“To what power must a beraised to obtain x?”a. f sxd 5 log2 x,x 5 32b. f sxd 5 log3 x,c. f sxd 5 log4 x,x52d. f sxd 5 log10 x,Solutiona. f s32d 5 log2 32 5 5b. f s1d 5 log3 1 5 0c. f s2d 5 log4 2 5 2111d. f s1005 22d 5 log10 100becausebecausebecausebecauseNow try Exercise 17.x511x 5 10025 5 32.30 5 1.41y2 5 !4 5 2.11022 5 101 2 5 100.
230Chapter 3Exponential and Logarithmic FunctionsThe logarithmic function with base 10 is called the common logarithmicfunction. It is denoted by log10 or simply by log. On most calculators, thisfunction is denoted by LOG . Example 2 shows how to use a calculator to evaluatecommon logarithmic functions. You will learn how to use a calculator to calculatelogarithms to any base in the next section.ExplorationComplete the table forf sxd 5 10 x.x2202112f sxdExample 2Evaluating Common Logarithms on a CalculatorUse a calculator to evaluate the function given by f sxd 5 log x at each value of x.Complete the table forf sxd 5 log x.x11001101b. x 5 13a. x 5 10c. x 5 2.5d. x 5 22Solution10100f sxdCompare the two tables. Whatis the relationship betweenf sxd 5 10 x and f sxd 5 log x?Function Valuea. f s10d 5 log 10b. f s13 d 5 log 13c. f s2.5d 5 log 2.5d. f s22d 5 logs22dGraphing Calculator KeystrokesLOG 10 ENTERx 1 4 3 cLOGENTERLOG 2.5 ENTERLOG x2 c 2 ENTERDisplay120.47712130.3979400ERRORNote that the calculator displays an error message (or a complex number) whenyou try to evaluate logs22d. The reason for this is that there is no real numberpower to which 10 can be raised to obtain 22.Now try Exercise 23.The following properties follow directly from the definition of the logarithmic function with base a.Properties of Logarithms1. loga 1 5 0 because a0 5 1.2. loga a 5 1 because a1 5 a.3. loga a x 5 x and a log a x 5 xInverse Properties4. If loga x 5 loga y, then x 5 y.One-to-One PropertyExample 3Using Properties of Logarithmsa. Simplify: log 4 1b. Simplify: log!7 !7c. Simplify: 6 log 620Solutiona. Using Property 1, it follows that log4 1 5 0.b. Using Property 2, you can conclude that log!7 !7 5 1.c. Using the Inverse Property (Property 3), it follows that 6 log 620 5 20.Now try Exercise 27.You can use the One-to-One Property (Property 4) to solve simple logarithmicequations, as shown in Example 4.
Section 3.2Example 4Logarithmic Functions and Their Graphs231Using the One-to-One Propertya. log3 x 5 log3 12Original equationx 5 12One-to-One Propertyb. logs2x 1 1d 5 log x 2x 1 1 5 x x 5 21c. log4sx2 2 6d 5 log4 10 x2 2 6 5 10 x2 5 16 x 5 4Now try Exercise 79.Graphs of Logarithmic FunctionsTo sketch the graph of y 5 loga x, you can use the fact that the graphs of inversefunctions are reflections of each other in the line y 5 x.Example 5Graphs of Exponential and Logarithmic FunctionsIn the same coordinate plane, sketch the graph of each function.a. f sxd 5 2xyf(x) 2 xSolutiona. For f sxd 5 2x, construct a table of values. By plotting these points and con-10y x8b. gsxd 5 log2 xnecting them with a smooth curve, you obtain the graph shown in Figure 3.13.6g(x) log 2 x4x21012314121248f sxd 5 2 x2x 2246810 2FIGURE223.13b. Because gsxd 5 log2 x is the inverse function of f sxd 5 2x, the graph of g isobtained by plotting the points s f sxd, xd and connecting them with a smoothcurve. The graph of g is a reflection of the graph of f in the line y 5 x, asshown in Figure 3.13.Now try Exercise 31.y54Example 63Sketch the graph of the common logarithmic function f sxd 5 log x. Identify thevertical asymptote.f(x) log x21Solutionx 11 2 3 4 5 6 7
218 Chapter 3 Exponential and Logarithmic Functions What you should learn Recognize and evaluate expo-nential functions with base a. Graph exponential functions and use the One-to-One Property. Recognize, evaluate, and graph exponential functions with base e. Use exponential
rational functions. O In Chapter 3, you will: Evaluate, analyze, and graph exponential and logarithmic functions. Apply properties of logarithms. Solve exponential and logarithmic equations. Model data using exponential, logarithmic, and logistic functions. O ENDANGERED SPECIES
Chapter 7 — Exponential and logarithmic functions . Exponential and logarithmic functions MB11 Qld-7 97 4 a 83 32 45 2 15 2 93 . MB11 Qld-7 98 Exponential and logarithmic functions c 53 152 32 53 (5
Unit 6 Exponential and Logarithmic Functions Lesson 1: Graphing Exponential Growth/Decay Function Lesson Goals: Identify transformations of exponential functions Identify the domain/range and key features of exponential functions Why do I need to Learn This? Many real life applications involve exponential functions.
6.3 Logarithmic Functions 6.7 Exponential and Logarithmic Models 6.4 Graphs of Logarithmic Functions 6.8 Fitting Exponential Models to Data Introduction . Solution When an amount grows at a !xed percent per unit time, the growth is exponential. To !nd A 0 we use the fact that A 0 is the amount at time zero, so A 0 10. To !nd k, use the fact .
324 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions 5.1 The Natural Logarithmic Function: Differentiation DEFINITION OF THE NATURAL LOGARITHMIC FUNCTION The natural logarithmic functionis defined by The domain of the natural logarithmic function is the set of all posi
348 Chapter 7 Exponential and Logarithmic Functions 7.1 Lesson WWhat You Will Learnhat You Will Learn Graph exponential growth and decay functions. Use exponential models to solve real-life problems. Exponential Growth and Decay Functions An exponential function has the form y abx, where
The Natural Logarithmic and Exponential The Natural Logarithmic and Exponential and Exponential Function FunctionFunctions sss: . Differentiate and integrate exponential functions that have bases other than e. Use exponential functions to model compound interest and exponential
Chapter 3 Exponential & Logarithmic Functions Section 3.1 Exponential Functions & Their Graphs Definition of Exponential Function: The exponential function f with base a is denoted f(x) a x where a 0, a 1 and x is any real number. Example 1: Evaluating Exponential Expressions Use
