Summary, Review, and Test

45 Chapter Equations and Inequalities Chapter Summar Summar, Review, and Test DEFINITIONS AND CONCEPTS EXAMPLES. Eponential Functions a. The eponential function with base b is defined b f = b, where b 7 0 and b Z. E., p. 89 b. Characteristics of eponential functions and graphs for 0 6 b 6 and b 7 are shown in the bo on E., p. 90; page 9. E., p. 90 c. Transformations involving eponential functions are summarized in Table. on page 9. E. 4, p. 9; E. 5, p. 9 d. The natural eponential function is f = e. The irrational number e is called the natural base, where E. 6, p. 94 is the value that a + n e L.78. e approaches as n : q. n b e. Formulas for compound interest: After t ears, the balance, A, in an account with principal P and annual E. 7, p. 96 interest rate r (in decimal form) is given b one of the following formulas:. For compoundings per ear: A = Pa + r nt n n b. For continuous compounding: A = Pe rt.. Logarithmic Functions a. Definition of the logarithmic function: For 7 0 and b 7 0, b Z, = log is equivalent to b b =. E., p. 40; The function f = log b is the logarithmic function with base b. This function is the inverse function of the eponential function with base b. E., p. 40; E., p. 40 b. Graphs of logarithmic functions for b 7 and 0 6 b 6 are shown in Figure.8 on page 404. Characteristics E. 6, p. 404 of the graphs are summarized in the bo on page 405. c. Transformations involving logarithmic functions are summarized in Table.4 on page 405. Figures.9. pp. 405 406 d. The domain of a logarithmic function of the form f = log b is the set of all positive real numbers. E. 7, p. 406; The domain of f = log b g consists of all for which g 7 0. E. 0, p. 408 e. Common and natural logarithms: f = log means f = log 0 and is the common logarithmic E. 8, p. 407; function. f = ln means f = log e and is the natural logarithmic function. E. 9, p. 408; E., p. 409 f. Basic Logarithmic Properties Base b (b>0, b ) Base 0 (Common Logarithms) Base e (Natural Logarithms) log b = 0 log = 0 ln = 0 log b b = log 0 = ln e = E. 4, p. 40 log b b = log 0 = ln e = b log b = 0 log = e ln = E. 5, p. 40. Properties of Logarithms a. The Product Rule: log b MN = log b M + log b N E., p. 44 b. The Quotient Rule: log b a M N b = log E., p. 45 b M - log b N c. The Power Rule: log b M p = p log b M E., p. 46

Summar, Review, and Test 45 DEFINITIONS AND CONCEPTS d. The Change-of-Base Propert: The General Propert log b M = log a M log a b Introducing Common Logarithms log b M = log M log b Introducing Natural Logarithms log b M = ln M ln b EXAMPLES E. 7, p. 40; E. 8, p. 40 e. Properties for epanding logarithmic epressions are given in the bo on page 46. E. 4, p. 47 f. Properties for condensing logarithmic epressions are given in the bo on page 48. E. 5, p. 48; E. 6, p. 48.4 Eponential and Logarithmic Equations a. An eponential equation is an equation containing a variable in an eponent. Some eponential equations can be solved b epressing each side as a power of the same base: If b M = b N, then M = N. Details are in the bo on page 44. b. The procedure for using logarithms to solve eponential equations is given in the bo on page 45. The solution procedure involves isolating the eponential epression. Take the natural logarithm on both sides for bases other than 0 and take the common logarithm on both sides for base 0. Simplif using ln b = ln b or ln e = or log 0 =. E., p. 44 E., p. 45; E., p. 46; E. 4, p. 46; E. 5, p. 47 c. A logarithmic equation is an equation containing a variable in a logarithmic epression. Some logarithmic E. 6, p. 47; equations can be epressed in the form log b M = c. The definition of a logarithm is used to rewrite the equation E. 7, p. 48 in eponential form: b c = M. See the bo on page 47.When checking logarithmic equations, reject proposed solutions that produce the logarithm of a negative number or the logarithm of 0 in the original equation. d. Some logarithmic equations can be epressed in the form log b M = log b N. Use the one-to-one propert E. 8, p. 49 to rewrite the equation without logarithms: M = N. See the bo on page 49..5 Eponential Growth and Deca; Modeling Data a. Eponential growth and deca models are given b A = A 0 e kt in which t represents time, A 0 is the E., p. 47; amount present at t = 0, and A is the amount present at time t. If k 7 0, the model describes growth and E., p. 49 k is the growth rate. If k 6 0, the model describes deca and k is the deca rate. c b. The logistic growth model, given b A = describes situations in which growth is limited. = c + ae, E., p. 44 -bt is a horizontal asmptote for the graph, and growth, A, can never eceed c. c. Newton s Law of Cooling: The temperature, T, of a heated object at time t is given b E. 4, p. 44 T = C + T 0 - Ce kt, where C is the constant temperature of the surrounding medium, T 0 is the initial temperature of the heated object, and k is a negative constant. d. Scatter plots for eponential and logarithmic models are shown in Figure.4 on page 44. When using a graphing E. 5, p. 444; utilit to model data, the closer that the correlation coefficient, r, is to - or, the better the model fits the data. E. 6, p. 445 e. Epressing an Eponential Model in Base is equivalent to = ae ln b # e: = ab. E. 7, p. 446 Review Eercises. In Eercises 4, the graph of an eponential function is given. Select the function for each graph from the following options: f = 4, g = 4 -, h = -4 -, r = -4 - +.... 4. In Eercises 5 9, graph f and g in the same rectangular coordinate sstem. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asmptotes. Use the graphs to determine each function s domain and range. 5. f = and g = -

454 Chapter Eponential and Logarithmic Functions 6. f = and g = - 5. ln e 5 6. log 7. ln 7. f = and g = - e 8. f = A and g = A B - B 8. log 9. log log 8 8 000 9. f = e and Use the compound interest formulas to solve Eercises 0. 0. Suppose that ou have $5000 to invest. Which investment ields the greater return over 5 ears: 5.5% compounded semiannuall or 5.5% compounded monthl?. Suppose that ou have $4,000 to invest. Which investment ields the greater return over 0 ears: 7% compounded monthl or 6.85% compounded continuousl?. A cup of coffee is taken out of a microwave oven and placed in a room. The temperature, T, in degrees Fahrenheit, of the coffee after t minutes is modeled b the function T = 70 + 0e -0.04855t. The graph of the function is shown in the figure. 40 00 T g = e 0. Graph f = and g = log in the same rectangular coordinate sstem. Use the graphs to determine each function s domain and range.. Graph f = A B and g = log in the same rectangular coordinate sstem. Use the graphs to determine each function s domain and range. In Eercises 5, the graph of a logarithmic function is given. Select the function for each graph from the following options: f = log, g = log-, h = log -, r = + log -.... 60 0 80 40 0 0 0 0 T = 70 + 0e 0.04855t Use the graph to answer each of the following questions. a. What was the temperature of the coffee when it was first taken out of the microwave? b. What is a reasonable estimate of the temperature of the coffee after 0 minutes? Use our calculator to verif this estimate. c. What is the limit of the temperature to which the coffee will cool? What does this tell ou about the temperature of the room? In Eercises 5, write each equation in its equivalent eponential form.. = log 49 7 4. = log 4 5. log 8 = In Eercises 6 8, write each equation in its equivalent logarithmic form. 6. 6 = 6 7. b 4 = 65 8. = 874 In Eercises 9 9, evaluate each epression without using a calculator. If evaluation is not possible, state the reason. 9. log 4 64 0.. log -9 log 5 5. log. log 4. log 8 6 4 7 7 0 T = 70 40 50 60 t 4. 5. In Eercises 6 8, begin b graphing f = log. Then use transformations of this graph to graph the given function. What is the graph s -intercept? What is the vertical asmptote? Use the graphs to determine each function s domain and range. 6. g = log - 7. h = - + log 8. r = log - In Eercises 9 40, graph f and g in the same rectangular coordinate sstem. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asmptotes. Use the graphs to determine each function s domain and range. 9. f = log and g = -log + 40. f = ln and g = -ln In Eercises 4 4, find the domain of each logarithmic function. 4. f = log 8 + 5 4. f = log - 4. f = ln - In Eercises 44 46, use inverse properties of logarithms to simplif each epression. log 4 44. ln e 6 45. e ln 46. 0

47. On the Richter scale, the magnitude, of an earthquake of I intensit I is given b R = log, where I is the intensit I 0 0 of a barel felt zero-level earthquake. If the intensit of an earthquake is 000I 0, what is its magnitude on the Richter scale? 48. Students in a pscholog class took a final eamination. As part of an eperiment to see how much of the course content the remembered over time, the took equivalent forms of the eam in monthl intervals thereafter. The average score, ft, for the group after t months is modeled b the function ft = 76-8 logt +, where 0 t. a. What was the average score when the eam was first given? b. What was the average score after months? 4 months? 6 months? 8 months? one ear? c. Use the results from parts (a) and (b) to graph f. Describe what the shape of the graph indicates in terms of the material retained b the students. 49. The formula t = c lna A A - N b describes the time, t, in weeks, that it takes to achieve master of a portion of a task. In the formula, A represents maimum learning possible, N is the portion of the learning that is to be achieved, and c is a constant used to measure an individual s learning stle. A 50-ear-old man decides to start running as a wa to maintain good health. He feels that the maimum rate he could ever hope to achieve is miles per hour. How man weeks will it take before the man can run 5 miles per hour if c = 0.06 for this person?. In Eercises 50 5, use properties of logarithms to epand each logarithmic epression as much as possible. Where possible, evaluate logarithmic epressions without using a calculator. 50. log 66 5. log 4 64 5. log 5. ln 64 A In Eercises 54 57, use properties of logarithms to condense each logarithmic epression. Write the epression as a single logarithm whose coefficient is. 54. log b 7 + log b 55. log - log 56. ln + 4 ln 57. ln - ln In Eercises 58 59, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. 58. log 6 7,48 59. log 4 0.86 R,.4 e Summar, Review, and Test 455 In Eercises 64 7, solve each eponential equation. Where necessar, epress the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approimation, correct to two decimal places, for the solution. 64. 4 - = 64 65. 5 = 5 66. 0 = 7000 67. 9 + = 7-68. 8 =,4 69. 9e 5 = 69 70. e - 5-7 = 7. 4 + 5 = 7,500 7. + 4 = 7-7. e - e - 6 = 0 In Eercises 74 79, solve each logarithmic equation. 74. log 4-5 = 75. 76. 77. 78. + 4 ln = 5 log + + log - = 4 log - - log + = ln + 4 - ln + = ln 79. log 4 + = log 4 - + log 4 + 5 80. The function P = 4.7e -0. models the average atmospheric pressure, P, in pounds per square inch, at an altitude of miles above sea level. The atmospheric pressure at the peak of Mt. Everest, the world s highest mountain, is 4.6 pounds per square inch. How man miles above sea level, to the nearest tenth of a mile, is the peak of Mt. Everest? 8. The amount of carbon dioide in the atmosphere, measured in parts per million, has been increasing as a result of the burning of oil and coal. The buildup of gases and particles traps heat and raises the planet s temperature, a phenomenon called the greenhouse effect. Carbon dioide accounts for about half of the warming. The function ft = 64.005 t projects carbon dioide concentration, ft, in parts per million, t ears after 000. Using the projections given b the function, when will the carbon dioide concentration be double the preindustrial level of 80 parts per million? 8. The function W = 0.7 ln + 0.05 models the average walking speed, W, in feet per second, of residents in a cit whose population is thousand. Visitors to New York Cit frequentl feel the are moving too slowl to keep pace with New Yorkers average walking speed of.8 feet per second. What is the population of New York Cit? Round to the nearest thousand. 8. Use the formula for compound interest with n compoundings per ear to solve this problem. How long, to the nearest tenth of a ear, will it take $,500 to grow to $0,000 at 6.5% annual interest compounded quarterl? In Eercises 60 6, determine whether each equation is true or false. Where possible, show work to support our conclusion. If the statement is false, make the necessar change(s) to produce a true statement. 60. 6. ln ln = 0 log + 9 log + 9 - log + = log + 6. log 4 = 4 log 6. ln e = ln e Use the formula for continuous compounding to solve Eercises 84 85. 84. How long, to the nearest tenth of a ear, will it take $50,000 to triple in value at 7.5% annual interest compounded continuousl? 85. What interest rate, to the nearest percent, is required for an investment subject to continuous compounding to triple in 5 ears?

456 Chapter Eponential and Logarithmic Functions.5 86. According to the U.S. Bureau of the Census, in 990 there were.4 million residents of Hispanic origin living in the United States. B 000, the number had increased to 5. million. The eponential growth function A =.4e kt describes the U.S. Hispanic population, A, in millions, t ears after 990. a. Find k, correct to three decimal places. b. Use the resulting model to project the Hispanic resident population in 00. c. In which ear will the Hispanic resident population reach 60 million? 87. Use the eponential deca model, A = A 0 e kt, to solve this eercise. The half-life of polonium-0 is 40 das. How long will it take for a sample of this substance to deca to 0% of its original amount? 88. The function 500,000 ft = + 499e -0.9t models the number of people, ft, in a cit who have become ill with influenza t weeks after its initial outbreak. a. How man people became ill with the flu when the epidemic began? b. How man people were ill b the end of the sith week? c. What is the limiting size of ft, the population that becomes ill? 89. Use Newton s Law of Cooling, T = C + T 0 - Ce kt, to solve this eercise. You are served a cup of coffee that has a temperature of 85 F. The room temperature is 65 F. After minutes, the temperature of the coffee is 55 F. a. Write a model for the temperature of the coffee, T, after t minutes. b. When will the temperature of the coffee be 05 F? Eercises 90 9 present data in the form of tables. For each data set shown b the table, 90. a. Create a scatter plot for the data. b. Use the scatter plot to determine whether an eponential function or a logarithmic function is the better choice for modeling the data. Growth of the Human Brain Age Percentage of Adult Size Brain 0% 50% 4 78% 6 88% 8 9% 0 95% 99% 9. In Eercises 9 9, rewrite the equation in terms of base e. Epress the answer in terms of a natural logarithm and then round to three decimal places. 9. = 7.6 9. = 6.50.4 94. The figure shows world population projections through the ear 50. The data are from the United Nations Famil Planning Program and are based on optimistic or pessimistic epectations for successful control of human population growth. Suppose that ou are interested in modeling these data using eponential, logarithmic, linear, and quadratic functions. Which function would ou use to model each of the projections? Eplain our choices. For the choice corresponding to a quadratic model, would our formula involve one with a positive or negative leading coefficient? Eplain. Population (billions) Tet Messaging in the U.S. Year 0 5 0 5 0 5 Projections in World Population Growth 6 billion High projection Medium projection Low projection 000 05 050 075 00 5 50 Year Source: U.N. Monthl Tet Messages (billions) 00 0.9 00.0 004. 005 8. 006 4. 007 8.9 Source: CTIA 0 billion billion 4 billion Source: Gerrig and Zimbardo, Pscholog and Life, Eighteenth Edition, Alln and Bacon, 008

Summar, Review, and Test 457 Chapter Test. Graph f = and g = + in the same rectangular coordinate sstem.. Graph f = log and g = log - in the same rectangular coordinate sstem.. Write in eponential form: log 5 5 =. 4. Write in logarithmic form: 6 = 6. 5. Find the domain: f = ln -. In Eercises 6 7, use properties of logarithms to epand each logarithmic epression as much as possible. Where possible, evaluate logarithmic epressions without using a calculator. 6. log 464 5 7. log 8 In Eercises 8 9, write each epression as a single logarithm. 8. 6 log + log 9. ln 7 - ln 0. Use a calculator to evaluate log 5 7 to four decimal places. In Eercises 8, solve each equation.. - = 9 + 4. 5 =.4. 400e 0.005 = 600 4. e - 6e + 5 = 0 5. log 6 4 - = 6. ln = 8 7. 8. log + log + 5 = ln - 4 - ln + = ln 6 9. On the decibel scale, the loudness of a sound, D, in decibels, I is given b D = 0 log, where I is the intensit of the I 0 sound, in watts per meter, and I 0 is the intensit of a sound barel audible to the human ear. If the intensit of a sound is 0 I 0, what is its loudness in decibels? (Such a sound is potentiall damaging to the ear.) In Eercises 0, simplif each epression. 0. ln e 5. log b b. log 6 Use the compound interest formulas to solve Eercises 5.. Suppose ou have $000 to invest. Which investment ields the greater return over 0 ears: 6.5% compounded semiannuall or 6% compounded continuousl? How much more (to the nearest dollar) is ielded b the better investment? 4. How long, to the nearest tenth of a ear, will it take $4000 to grow to $8000 at 5% annual interest compounded quarterl? 5. What interest rate, to the nearest tenth of a percent, is required for an investment subject to continuous compounding to double in 0 ears? 6. The function A = 8.4e -0.00t models the population of German, A, in millions, t ears after 006. a. What was the population of German in 006? b. Is the population of German increasing or decreasing? Eplain. c. In which ear will the population of German be 8.5 million? 7. The 990 population of Europe was 509 million; in 000, it was 79 million. Write the eponential growth function that describes the population of Europe, in millions, t ears after 990. 8. Use the eponential deca model, A = A 0 e kt, to solve this eercise. The half-life of iodine- is 7. das. How long will it take for a sample of this substance to deca to 0% of its original amount? Round to one decimal place. 9. The logistic growth function 40 ft = + 9e -0.65t describes the population, ft, of an endangered species of elk t ears after the were introduced to a nonthreatening habitat. a. How man elk were initiall introduced to the habitat? b. How man elk are epected in the habitat after 0 ears? c. What is the limiting size of the elk population that the habitat will sustain? In Eercises 0, determine whether the values in each table belong to an eponential function, a logarithmic function, a linear function, or a quadratic function. 0. 0 - - 4-5.. 0 5 5 5 4 65. - 0 9 7 0 0 4 4. Rewrite = 960.8 in terms of base e. Epress the answer in terms of a natural logarithm and then round to three decimal places.

458 Chapter Eponential and Logarithmic Functions Cumulative Review Eercises (Chapters P ) In Eercises 8, solve each equation or inequalit.. ƒ - 4 ƒ =. + + 5 = 0. 4 + - - + = 0 4. e 5 - = 96 5. log + 5 + log - = 4 6. ln + 4 + ln + = ln + 7. 4-5 Ú -6 8. ƒ - 4 ƒ In Eercises 9 4, graph each equation in a rectangular coordinate sstem. If two functions are indicated, graph both in the same sstem. 9. - + + = 4 0. f = - -. f = - - 4. f = - +. f = - 4 and f - 4. f = ln and g = ln - + 5. Write the point-slope form and the slope-intercept form of the line passing through (, ) and, -. 6. If f = and g = +, find f g and g f. 7. You discover that the number of hours ou sleep each night varies inversel as the square of the number of cups of coffee consumed during the earl evening. If cups of coffee are consumed, ou get 8 hours of sleep. If the number of cups of coffee is doubled, how man hours should ou epect to sleep? A baseball plaer hits a pop fl into the air. The function st = -6t + 64t + 5 models the ball s height above the ground, st, in feet, t seconds after it is hit. Use the function to solve Eercises 8 9. 8. When does the baseball reach its maimum height? What is that height? 9. After how man seconds does the baseball hit the ground? Round to the nearest tenth of a second. 0. You are paid time-and-a-half for each hour worked over 40 hours a week. Last week ou worked 50 hours and earned $660. What is our normal hourl salar?