Chapter Summary. What did you learn? 270 Chapter 3 Exponential and Logarithmic Functions

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1 0_00R.qd /7/05 0: AM Page Chapter Eponential and Logarithmic Functions Chapter Summar What did ou learn? Section. Review Eercises Recognize and evaluate eponential functions with base a (p. ). Graph eponential functions and use the One-to-One Propert (p. 9). 7 Recognize, evaluate, and graph eponential functions with base e (p. ). 7 Use eponential functions to model and solve real-life problems (p. ). 5 0 Section. Recognize and evaluate logarithmic functions with base a (p. 9). 5 Graph logarithmic functions (p. ). 5 5 Recognize, evaluate, and graph natural logarithmic functions (p. ). 59 Use logarithmic functions to model and solve real-life problems (p. 5). 9,70 Section. Use the change-of-base formula to rewrite and evaluate logarithmic epressions (p. 9). 7 7 Use properties of logarithms to evaluate or rewrite logarithmic epressions (p. 0) Use properties of logarithms to epand or condense logarithmic epressions (p. ) Use logarithmic functions to model and solve real-life problems (p. ). 95,9 Section. Solve simple eponential and logarithmic equations (p. ) Solve more complicated eponential equations (p. 7). 05 Solve more complicated logarithmic equations (p. 9). 9 Use eponential and logarithmic equations to model and solve 5, real-life problems (p. 5). Section.5 Recognize the five most common tpes of models involving eponential 7 and logarithmic functions (p. 57). Use eponential growth and deca functions to model and solve real-life problems (p. 5). Use Gaussian functions to model and solve real-life problems (p. ). 9 Use logistic growth functions to model and solve real-life problems (p. ). 50 Use logarithmic functions to model and solve real-life problems (p. ). 5, 5

2 0_00R.qd /7/05 0:5 AM Page 7 Review Eercises 7 Review Eercises. In Eercises, evaluate the function at the indicated value of. Round our result to three decimal places. Function Value f. f 0 f 0.5 f 7 5 f 7 0. f In Eercises 7 0, match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) 5 5 (b) (d) 7. f. f 9. f 0. f In Eercises, use the graph of transformation that ields the graph of. f 5,. f,.. f, f, g 5 g g g to describe the In Eercises 5, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function. 5. f. f 7. f.5. f.5 f g f 5 0. f 5. f. In Eercises, use the One-to-One Propert to solve the equation for e 5 7 e 5. e e In Eercises 7 0, evaluate the function given b f e at the indicated value of. Round our result to three decimal places In Eercises, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function.. h e. h e. f e. s t e t, t > 0 Compound Interest In Eercises 5 and, complete the table to determine the balance A for P dollars invested at rate r for t ears and compounded n times per ear. 5. P $500, r.5%, t 0 ears. P $000, r 5%, t 0 ears f 5 n 5 Continuous A 7. Waiting Times The average time between incoming calls at a switchboard is minutes. The probabilit F of waiting less than t minutes until the net incoming call is approimated b the model F t e t. A call has just come in. Find the probabilit that the net call will be within (a) minute. (b) minutes. (c) 5 minutes.. Depreciation After t ears, the value V of a car that originall cost $,000 is given b V t,000 t. (a) Use a graphing utilit to graph the function. (b) Find the value of the car ears after it was purchased. (c) According to the model, when does the car depreciate most rapidl? Is this realistic? Eplain.

3 0_00R.qd /7/05 :9 PM Page 7 7 Chapter Eponential and Logarithmic Functions 9. Trust Fund On the da a person is born, a deposit of $50,000 is made in a trust fund that pas.75% interest, compounded continuousl. (a) Find the balance on the person s 5th birthda. (b) How much longer would the person have to wait for the balance in the trust fund to double? 0. Radioactive Deca Let Q represent a mass of plutonium Pu (in grams), whose half-life is. ears. The quantit of plutonium present after t ears is given b Q 00 t.. (a) Determine the initial quantit (when t 0). (b) Determine the quantit present after 0 ears. (c) Sketch the graph of this function over the interval t 0 to t 00.. In Eercises, write the eponential equation in logarithmic form e e 0 In Eercises 5, evaluate the function at the indicated value of without using a calculator. Function 5. f log. g log 9 7. g log. f log Value 000 In Eercises 9 5, use the One-to-One Propert to solve the equation for. 9. log 7 log 50. log 0 log 5 5. ln 9 ln 5. ln ln In Eercises 5 5, find the domain, -intercept, and vertical asmptote of the logarithmic function and sketch its graph. 5. g log 7 5. g log f log 5. f log 57. f log 5 5. f log In Eercises 59, use a calculator to evaluate the function given b f ln at the indicated value of. Round our result to three decimal places if necessar e. e In Eercises 5, find the domain, -intercept, and vertical asmptote of the logarithmic function and sketch its graph. 5. f ln. f ln 7. h ln. f ln 9. Antler Spread The antler spread a (in inches) and shoulder height h (in inches) of an adult male American elk are related b the model h log a 0 7. Approimate the shoulder height of a male American elk with an antler spread of 55 inches. 70. Snow Removal The number of miles s of roads cleared of snow is approimated b the model s 5 ln h, ln h 5 where h is the depth of the snow in inches. Use this model to find s when h 0 inches.. In Eercises 7 7, evaluate the logarithm using the change-of-base formula. Do each eercise twice, once with common logarithms and once with natural logarithms. Round our the results to three decimal places. 7. log 9 7. log log 5 7. log 0. In Eercises 75 7, use the properties of logarithms to rewrite and simplif the logarithmic epression. 75. log 7. log 77. ln 0 7. ln e In Eercises 79, use the properties of logarithms to epand the epression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) 79. log log 7. log. log 7. ln z. ln 5. ln. ln, > In Eercises 7 9, condense the epression to the logarithm of a single quantit. 7. log 5 log. log log z 9. ln ln 90. ln ln 9. log 7 log 9. log 5 log 9. ln ln 9. 5 ln ln ln

4 0_00R.qd /7/05 0:5 AM Page 7 Review Eercises Climb Rate The time t (in minutes) for a small plane to climb to an altitude of h feet is modeled b,000 t 50 log,000 h where,000 feet is the plane s absolute ceiling. (a) Determine the domain of the function in the contet of the problem. (b) Use a graphing utilit to graph the function and identif an asmptotes. (c) As the plane approaches its absolute ceiling, what can be said about the time required to increase its altitude? (d) Find the time for the plane to climb to an altitude of 000 feet. 9. Human Memor Model Students in a learning theor stud were given an eam and then retested monthl for months with an equivalent eam. The data obtained in the stud are given as the ordered pairs t, s, where t is the time in months after the initial eam and s is the average score for the class. Use these data to find a logarithmic equation that relates t and s.,.,, 7.,, 7.,,.5, 5, 7.,, 5.. In Eercises 97 0, solve for e 00. e 0. log 0. log 0. ln 0. ln In Eercises 05, solve the eponential equation algebraicall. Approimate our result to three decimal places. 05. e 0. e e e 0. e e 7e 0 0. e e 0 In Eercises 5, use a graphing utilit to graph and solve the equation. Approimate the result to three decimal places e 0.. e. 9 In Eercises 9 0, solve the logarithmic equation algebraicall. Approimate the result to three decimal places. 9. ln. 0. ln ln 5. ln 5. ln ln. ln 5. ln. ln ln 5 7. log log log. log log log 5 9. log 0. log In Eercises, use a graphing utilit to graph and solve the equation. Approimate the result to three decimal places.. ln. log 0. ln 5 0. log 0 5. Compound Interest You deposit $7550 in an account that pas 7.5% interest, compounded continuousl. How long will it take for the mone to triple?. Meteorolog The speed of the wind S (in miles per hour) near the center of a tornado and the distance d (in miles) the tornado travels are related b the model S 9 log d 5. On March, 95, a large tornado struck portions of Missouri, Illinois, and Indiana with a wind speed at the center of about miles per hour. Approimate the distance traveled b this tornado..5 In Eercises 7, match the function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (c) (e) 5 (b) (d) (f) 0

5 0_00R.qd /7/05 0:5 AM Page 7 7 Chapter Eponential and Logarithmic Functions 7. e. e 9. ln 0. 7 log. e. e In Eercises and, find the eponential model ae b that passes through the points.. 0,,,. 0,, 5, 5 5. Population The population P of South Carolina (in thousands) from 990 through 00 can be modeled b P 99e 0.05t, where t represents the ear, with t 0 corresponding to 990. According to this model, when will the population reach.5 million? (Source: U.S. Census Bureau). Radioactive Deca The half-life of radioactive uranium II U is about 50,000 ears. What percent of a present amount of radioactive uranium II will remain after 5000 ears? 7. Compound Interest A deposit of $0,000 is made in a savings account for which the interest is compounded continuousl. The balance will double in 5 ears. (a) What is the annual interest rate for this account? (b) Find the balance after ear.. Wildlife Population A species of bat is in danger of becoming etinct. Five ears ago, the total population of the species was 000. Two ears ago, the total population of the species was 00. What was the total population of the species one ear ago? 9. Test Scores The test scores for a biolog test follow a normal distribution modeled b 0.099e 7, where is the test score. (a) Use a graphing utilit to graph the equation. (b) From the graph in part (a), estimate the average test score. 50. Tping Speed In a tping class, the average number N of words per minute tped after t weeks of lessons was found to be N 57 5.e 0.t Find the time necessar to tpe (a) 50 words per minute and (b) 75 words per minute. 5. Sound Intensit The relationship between the number of decibels and the intensit of a sound I in watts per square centimeter is 0 log I Determine the intensit of a sound in watts per square centimeter if the decibel level is Geolog On the Richter scale, the magnitude R of an earthquake of intensit I is given b R log I I 0 where I 0 is the minimum intensit used for comparison. Find the intensit per unit of area for each value of R. (a) R. (b) R.5 (c) R 9. Snthesis True or False? In Eercises 5 and 5, determine whether the equation is true or false. Justif our answer. 5. log b b 5. ln ln ln 55. The graphs of e kt are shown where k a, b, c, and d. Which of the four values are negative? Which are positive? Eplain our reasoning. (a) (b) (c) (0, ) 0. (0, ) = e ct = e at (d) (0, ) (0, ) = e dt = e bt

6 0_00R.qd /7/05 0:5 AM Page 75 Chapter Test 75 Chapter Test Take this test as ou would take a test in class. When ou are finished, check our work against the answers given in the back of the book. In Eercises, evaluate the epression. Approimate our result to three decimal places e 7 0 e. In Eercises 5 7, construct a table of values. Then sketch the graph of the function. 5. f 0. f 7. f e. Evaluate (a) log and (b). ln e. In Eercises 9, construct a table of values. Then sketch the graph of the function. Identif an asmptotes. 9. f log 0. f ln. f ln In Eercises, evaluate the logarithm using the change-of-base formula. Round our result to three decimal places.. log 7. log log In Eercises 5 7, use the properties of logarithms to epand the epression as a sum, difference, and/or constant multiple of logarithms. 5. log. ln 5 a 7. log 7 z Eponential Growth,000 (9,,77) 0,000,000,000,000,000 (0, 75) 0 FIGURE FOR 7 t In Eercises 0, condense the epression to the logarithm of a single quantit.. log log 9. ln ln 0. ln ln 5 ln In Eercises, solve the equation algebraicall. Approimate our result to three decimal places e e. ln 5. ln 7. log log 5 7. Find an eponential growth model for the graph shown in the figure.. The half-life of radioactive actinium 7 Ac is.77 ears. What percent of a present amount of radioactive actinium will remain after 9 ears? 9. A model that can be used for predicting the height H (in centimeters) of a child based on his or her age is H ln,, where is the age of the child in ears. (Source: Snapshots of Applications in Mathematics) (a) Construct a table of values. Then sketch the graph of the model. (b) Use the graph from part (a) to estimate the height of a four-ear-old child. Then calculate the actual height using the model.

7 0_00R.qd /7/05 0:5 AM Page 7 7 Chapter Eponential and Logarithmic Functions Cumulative Test for Chapters FIGURE FOR Take this test to review the material from earlier chapters. When ou are finished, check our work against the answers given in the back of the book.. Plot the points, and,. Find the coordinates of the midpoint of the line segment joining the points and the distance between the points. In Eercises, graph the equation without using a graphing utilit Find an equation of the line passing through and,.. Eplain wh the graph at the left does not represent as a function of. 7. Evaluate (if possible) the function given b f for each value. (a) f (b) f (c) f s. Compare the graph of each function with the graph of. (Note: It is not necessar to sketch the graphs.) (a) r (b) h (c) g In Eercises 9 and 0, find (a) f g, (b) f g, (c) fg, and (d) f/g. What is the domain of f/g? 9. f, g 0. f, g In Eercises and, find (a) f g and (b) g f. Find the domain of each composite function.. f, g. f,. Determine whether h 5 has an inverse function. If so, find the inverse function.. The power P produced b a wind turbine is proportional to the cube of the wind speed S. A wind speed of 7 miles per hour produces a power output of 750 kilowatts. Find the output for a wind speed of 0 miles per hour. 5. Find the quadratic function whose graph has a verte at, 5 and passes through the point, 7. In Eercises, sketch the graph of the function without the aid of a graphing utilit.. h 7. f t t t. g s s s 0 In Eercises 9, find all the zeros of the function and write the function as a product of linear factors. 9. f 0. f. f 0 0, g

8 0_00R.qd /7/05 0:5 AM Page 77 Cumulative Test for Chapters 77. Use long division to divide b.. Use snthetic division to divide 5 b.. Use the Intermediate Value Theorem and a graphing utilit to find intervals one unit in length in which the function g is guaranteed to have a zero. Approimate the real zeros of the function. In Eercises 5 7, sketch the graph of the rational function b hand. Be sure to identif all intercepts and asmptotes. 5. f f f In Eercises and 9, solve the inequalit. Sketch the solution set on the real number line In Eercises 0 and, use the graph of f to describe the transformation that ields the graph of g. 0. f 5, g 5. f., g. In Eercises 5, use a calculator to evaluate the epression. Round our result to three decimal places.. log 9. log 7. ln 5. ln 0 5. Use the properties of logarithms to epand ln where >., 7. Write ln ln 5 as a logarithm of a single quantit. In Eercises 0, solve the equation algebraiciall. Approimate the result to three decimal places. Year TABLE FOR Sales, S e 7 9. e e 0 0. ln. The sales S (in billions of dollars) of lotter tickets in the United States from 997 through 00 are shown in the table. (Source: TLF Publications, Inc.) (a) Use a graphing utilit to create a scatter plot of the data. Let t represent the ear, with t 7 corresponding to 997. (b) Use the regression feature of the graphing utilit to find a quadratic model for the data. (c) Use the graphing utilit to graph the model in the same viewing window used for the scatter plot. How well does the model fit the data? (d) Use the model to predict the sales of lotter tickets in 00. Does our answer seem reasonable? Eplain.. The number N of bacteria in a culture is given b the model N 75e kt, where t is the time in hours. If N 0 when t, estimate the time required for the population to double in size.

9 0_00R.qd /7/05 :9 PM Page 7 Proofs in Mathematics Each of the following three properties of logarithms can be proved b using properties of eponential functions. Slide Rules The slide rule was invented b William Oughtred (57 0) in 5. The slide rule is a computational device with a sliding portion and a fied portion. A slide rule enables ou to perform multiplication b using the Product Propert of Logarithms. There are other slide rules that allow for the calculation of roots and trigonometric functions. Slide rules were used b mathematicians and engineers until the invention of the hand-held calculator in 97. Properties of Logarithms (p. 0) Let a be a positive number such that a, and let n be a real number. If u and v are positive real numbers, the following properties are true.. Product Propert: Proof Let log a u and The corresponding eponential forms of these two equations are a u and To prove the Product Propert, multipl u and v to obtain uv a a a. The corresponding logarithmic form of uv a is log a uv. So, log a uv log a u log a v. To prove the Quotient Propert, divide u b v to obtain u a v a a. a v. Logarithm with Base a log a uv log a u log a v log a v. Natural Logarithm ln uv ln u ln v. Quotient Propert: log u ln u a ln u ln v v log a u log a v v. Power Propert: log ln u n a u n n log a u n ln u The corresponding logarithmic form of u v a is log a u v. So, log u a v log a u log a v. To prove the Power Propert, substitute a for u in the epression log a u n, as follows. log a u n log a a n Substitute for u. log a a n Propert of eponents n Inverse Propert of Logarithms n log a u Substitute log a u for. So, log a u n n log a u. a 7

10 0_00R.qd /7/05 0:5 AM Page 79 P.S. Problem Solving This collection of thought-provoking and challenging eercises further eplores and epands upon concepts learned in this chapter.. Graph the eponential function given b a for a 0.5,., and.0. Which of these curves intersects the line? Determine all positive numbers a for which the curve a intersects the line.. Use a graphing utilit to graph e and each of the functions and 5.,,, Which function increases at the greatest rate as approaches?. Use the result of Eercise to make a conjecture about the rate of growth of and n e, where n is a natural number and approaches.. Use the results of Eercises and to describe what is implied when it is stated that a quantit is growing eponentiall. 5. Given the eponential function f a show that (a) f u v f u f v. (b) f f.. Given that and g e e f e e show that f g. 7. Use a graphing utilit to compare the graph of the function given b e with the graph of each given function. n! (read n factorial is defined as n!... n n. (a) (b)!! (c)!!!. Identif the pattern of successive polnomials given in Eercise 7. Etend the pattern one more term and compare the graph of the resulting polnomial function with the graph of e. What do ou think this pattern implies? 9. Graph the function given b f e e.! From the graph, the function appears to be one-to-one. Assuming that the function has an inverse function, find f. 0. Find a pattern for f if f a a where a > 0, a.. B observation, identif the equation that corresponds to the graph. Eplain our reasoning. (a) e (b) e (c) e. You have two options for investing $500. The first earns 7% compounded annuall and the second earns 7% simple interest. The figure shows the growth of each investment over a 0-ear period. (a) Identif which graph represents each tpe of investment. Eplain our reasoning. Investment (in dollars) Year (b) Verif our answer in part (a) b finding the equations that model the investment growth and graphing the models. (c) Which option would ou choose? Eplain our reasoning.. Two different samples of radioactive isotopes are decaing. The isotopes have initial amounts of c and c, as well as half-lives of k and k, respectivel. Find the time required for the samples to deca to equal amounts. t 79

11 0_00R.qd /7/05 0:5 AM Page 0. A lab culture initiall contains 500 bacteria. Two hours later, the number of bacteria has decreased to 00. Find the eponential deca model of the form B B 0 a kt that can be used to approimate the number of bacteria after t hours. 5. The table shows the colonial population estimates of the American colonies from 700 to 70. (Source: U.S. Census Bureau) Year Population ,900 70,700 70, , ,00 750,70,00 70,59,00 770,,00 70,70,00 In each of the following, let represent the population in the ear t, with t 0 corresponding to 700. (a) Use the regression feature of a graphing utilit to find an eponential model for the data. (b) Use the regression feature of the graphing utilit to find a quadratic model for the data. (c) Use the graphing utilit to plot the data and the models from parts (a) and (b) in the same viewing window. (d) Which model is a better fit for the data? Would ou use this model to predict the population of the United States in 00? Eplain our reasoning. log. Show that a log a b log a b. 7. Solve ln ln.. Use a graphing utilit to compare the graph of the function given b ln with the graph of each given function. (a) (b) (c) 9. Identif the pattern of successive polnomials given in Eercise. Etend the pattern one more term and compare the graph of the resulting polnomial function with the graph of ln. What do ou think the pattern implies? 0. Using ab and take the natural logarithm of each side of each equation. What are the slope and -intercept of the line relating and ln for ab? What are the slope and -intercept of the line relating ln and ln for a b? In Eercises and, use the model 0. ln, a b which approimates the minimum required ventilation rate in terms of the air space per child in a public school classroom. In the model, is the air space per child in cubic feet and is the ventilation rate per child in cubic feet per minute.. Use a graphing utilit to graph the model and approimate the required ventilation rate if there is 00 cubic feet of air space per child.. A classroom is designed for 0 students. The air conditioning sstem in the room has the capacit of moving 50 cubic feet of air per minute. (a) Determine the ventilation rate per child, assuming that the room is filled to capacit. (b) Estimate the air space required per child. (c) Determine the minimum number of square feet of floor space required for the room if the ceiling height is 0 feet. In Eercises, (a) use a graphing utilit to create a scatter plot of the data, (b) decide whether the data could best be modeled b a linear model, an eponential model, or a logarithmic model, (c) eplain wh ou chose the model ou did in part (b), (d) use the regression feature of a graphing utilit to find the model ou chose in part (b) for the data and graph the model with the scatter plot, and (e) determine how well the model ou chose fits the data..,.0,.5,.5,,.0,, 5.,, 7.0,, 7..,.,.5,.7,, 5.5,, 9.9,,.,,.0 5., 7.5,.5, 7.0,,.,, 5.0,,.5,,.0., 5.0,.5,.0,,.,, 7.,,.,, 9.0 0