8. Practice A For use with pages 65 7 Match the function with its graph.. f. f.. f 5. f 6. f f Lesson 8. A. B. C. (, 6) (0, ) (, ) (0, ) ( 0, ) (, ) D. E. F. (0, ) (, 6) ( 0, ) (, ) (, ) (0, ) Eplain how the graph of g can be obtained from the graph of f. 7. f 8. f g 0. f 5. f. g 5 g 5 g Identif the -intercept and asmptote of the graph of the function... 6 5. 6. 7. 8. 5 f 5 g 5 f g Account Balance You deposit $000 in an account that earns 5% annual interest. Find the balance after ear if the interest is compounded with the given frequenc. a. annuall b. quarterl c. monthl Algebra
8. Practice B For use with pages 65 7 Match the function with its graph... f f.. f 5. f 6. A. B. C. f f (, ) ( 0, ) (, ) (0, ) (, 0) ( 0, 7 9) Lesson 8. D. E. F. (, ) (0, ) ( 0, ) (, ) ( 0, ) (, ) Eplain how the graph of g can be obtained from the graph of f. 7. f 8. f 0 g g 0 Identif the -intercept and the asmptote of the graph of the function. f g 0... Graph the function... 5. 6. 7. 8. 0.. Computer Usage In Eercises, use the following information. From 99 through 995, the number of computers per 00 people worldwide can be modeled b C 5..5 t where t is the number of ears since 9. Identif the initial amount, the growth factor, and the annual percent increase.. Graph the function.. Estimate the number of computers per 000 people worldwide in 000. Algebra 5
8. Practice A For use with pages 7 79 Tell whether the function represents eponential growth or eponential deca... f 5 f.. 5. f f 0.7 6. Match the function with its graph. 7. 8. 0... A. B. C. (0, ) (, ) ( 0, ) (, 6) f 6 f (0, ) (, ) Lesson 8. D. E. F. (0, ) (, ) (0, ) (, ) ( 0, ) (, 6) Identif the -intercept and asmptote of the graph of the function... 0. 5. 6. 7. 5 8. 8 9 Radioactive Deca Ten grams of Carbon is stored in a container. The amount C (in grams) of Carbon present after t ears can be modeled b C 0 0.99987 t. How much Carbon is present after 000 ears? 5 Algebra 9
8. Practice B For use with pages 7 79 Tell whether the function represents eponential growth or eponential deca. f 5 7 f 7 5... f Match the function with its graph. 5. 5. 6. 5 5 5 7. 8. A. B. C. 5 5 (0, 5) Lesson 8. (, ) D. E. F. (, ) ( 0, 5) (, 5) (0, ) (, 5) 5 (, ) (, ) ( 0, 5 ) (0, ) (0, 5) Graph the function. 0... 5 5.. 5. Value of the Dollar In Eercises 6 8, use the following information. From 990 through 998, the value of the dollar has been shrinking. That is, ou cannot bu as much with a dollar toda as ou could in 990. The shrinking value can be modeled b V. 0.97 t, where t is the number of ears since 990. 6. How much was a 998 dollar worth in 99? 7. Graph the model. 8. Estimate the ear in which the 998 dollar was worth $.07. 0 Algebra
8. Practice B For use with pages 80 85 Use a calculator to evaluate the epression. Round the result to three decimal places. e 5 e e..... e Tell whether the function is an eample of eponential growth or eponential deca. 5. f e 6. f e 7. 8. f f 5 e5 e 0. f e f e 5 Simplif the epression. e. e 5.. e. e 5. e e 6. e e 7. 6e 8. e e Complete the table of values. Round to two decimal places. 0. f e..5 0.5 f() f e e e e.5 0.5 f(). f e. f e.5 0.5.5 0.5 f() f() Lesson 8. Graph the function and identif the horizontal asmptote.. f e 5. f e 6. 7. f e 8. f e Interest In Eercises 0, use the following information. You deposit $00 in an account that pas 5% annual interest. After 0 ears, ou withdraw the mone. 0. Find the balance in the account if the interest was compounded quarterl.. Find the balance in the account if the interest was compounded continuousl.. Which tpe of compounding ielded the greatest balance? f e f e.5 Algebra
8. Practice C For use with pages 80 85 Use a calculator to evaluate the epression. Round the result to three decimal places. e e.6.... e e e Simplif the epression. e 5. e e 6. 7. 8. e 0. e 6 Identif the horizontal asmptote of the function.. f e. f e. Graph the function. State the domain and range.. f e 5. f e 6. 7. f e 5 8. f e e e 8e f 5e 0.0 f e f 5 e Carbon Dating In Eercises 0, use the following information. Carbon dating is a process to estimate the age of organic material. In carbon dating the formula used is R e t 8 0 where R is the ratio of Carbon to Carbon and t is time in ears. 0. Is the model an eample of eponential growth or eponential deca?. Graph the function.. Use the graph to estimate the age of a fossil whose Carbon to Carbon ratio is 0. Learning Curve In Eercises 6, use the following information. The management at a factor has determined that a worker can produce a maimum of 0 units per da. The model 0 0e 0.07t indicates the number of units that a new emploee can produce per da after t das on the job.. Is the model an eample of eponential growth or eponential deca?. Graph the function. 5. How man units can be produced per da b an emploee who has been on the job 8 das? 6. Use the graph to estimate how man das of emploment are required for a worker to produce 5 units per da. Lesson 8. Algebra
8. Practice A For use with pages 86 9 Rewrite the equation in eponential form.. log 8. log 5 5.. log 7 9 5. log 6 6. Evaluate the epression without using a calculator. 7. log 8. log 0. log 0 00. log 7. Use a calculator to evaluate the epression. Round the result to three decimal places.. log 6. log 0. 5. 6. ln 8 7. ln 0. 8. Simplif the epression. 7 log 7 0. 7 log 7.. log. log 5 5. Match the function with its graph. 5. f log 6. f log 5 7. A. B. C. log 7 log 6 6 log 8 6 log 8 8 log.7 ln 6. log log f log Match the function with the graph of its inverse. 8. f log f log 0. A. B. C. f ln Lesson 8.. Sound The level of sound V in decibels with an intensit I can be modeled b V 0 log I 0 6, where I is intensit in watts per centimeter. Loud music can have an intensit of centimeter. Find the level of sound of loud music. 56 Algebra 0 5 watts per
8. Practice B For use with pages 86 9 Rewrite the equation in eponential form.. log 6. log 8.. log 5. log 9 5 5 6. Use a calculator to evaluate the epression. Round the result to three decimal places. 7. ln 8. log.5 Evaluate the logarithm without using a calculator. 0. log 7. log.. log. log 5 5 8 5. Find the inverse of the function. 6. f log 7. f ln 8. f log 0. f log. Graph the function.. f log 6. f log 6. 5. f log 6 6. f log 6 7. log 0 log 8 ln log log 6 f log f log 6 f log 6 f log 6 8. Galloping Speed Four-legged animals run with two different tpes of motion: trotting and galloping. An animal that is trotting has at least one foot on the ground at all times. An animal that is galloping has all four feet off the ground at times. The number s of strides per minute at which an animal breaks from a trot to a gallop is related to the animal s weight w (in pounds) b the model S 56. 7.9 log w. Approimate the number of strides per minute for a 500 pound horse when it breaks from a trot to a gallop. Tornadoes The wind speed S (in miles per hour) near the center of a tornado is related to the distance d (in miles) the tornado travels b the model S 9 log d 65. Approimate the wind speed of a tornado that traveled 50 miles. Lesson 8. Algebra 57
Lesson 8.5 LESSON 8.5 Practice A For use with pages 9 99 Use the properties of logarithms to rewrite the epression in terms of log and log 7. Then use log 0.0 and log 7 0.85 to approimate the epression.. log. log.. 5. log 7 6. log 9 log 7 log 7 Epand the epression. 7. log 8. log 9 0.. log 5. log 6 6. log. log 5. Condense the epression. 6. log log 5 7. log log 7 8. log log 0. ln ln.. ln ln. log 5 log. Use the change-of-base formula to rewrite the epression. Then use a calculator to evaluate the epression. Round our result to three decimal places. 5. log 5 6. log 7 0 7. 8. log log 6 00 5 0. log 5 ln log 7 log log log log 6 log log 8 log 7 log 5 Investments In Eercises and, use the following information. You want to invest in a stock whose value has been increasing b approimatel 5% each ear. The time required for an initial investment of I 0 to grow to I can be modeled b t where I 0 ln I I 0 0.09, and I are measured in dollars and t is measured in ears.. Epand the epression for t.. Assume that ou have $000 to invest. Complete the table to show how long our investment would take to double, triple, and quadruple. I 000 000 000 t 70 Algebra
8.5 Practice B For use with pages 9 99 Use the properties of logarithms to rewrite the epression in terms of log and log. Then use log 0.77 and log 0.60 to approimate the epression... log. log 9. log 6 5. log 6. log 7 log Lesson 8.5 Epand the epression. 7. log 8. log 6 5 log 0. log. log z.. log. log 0 5. log 5 log z Condense the epression. 6. log 7 log 7. log 5 log 5 8. log 5 log log log log 0. log log. log log log 5 Use the change-of-base formula to rewrite the epression. Then use a calculator to evaluate the epression. Round our result to three decimal places if necessar.. log. log 6. 5. log 0.8 6. log.5.8 7. log 0.5 log 6 Henderson-Hasselbach Formula information. In Eercises 8, use the following The ph of a patient s blood can be calculated using the Henderson-Hasselbach Formula, ph 6. log B C, where B is the concentration of bicarbonate and C is the concentration of carbonic acid. The normal ph of blood is approimatel 7.. 8. Epand the right side of the formula. A patient has a bicarbonate concentration of and a carbonic acid concentration of. Find the ph of the patient s blood. 0. Is the patient s ph in Eercise 9 below normal or above normal?. A patient has a bicarbonate concentration of. Graph the model.. Use the graph to approimate the concentration of carbonic acid required for the patient to have normal blood ph. Algebra 7
8.6 Practice A For use with pages 50 508 Tell whether the -value is a solution of the equation.. ln 9, e 9. ln, e.. ln 8, e 8 5. ln 6, e 6 6. ln 7, 7 e ln, e Tell whether the -value is a solution of the equation. 7. e 5, 5 8. e 7, ln 7 0. e 8, ln. e,. Solve the equation... 5 5. 6. e e 7 7. e e 8. e, log 5e 7, ln 0 0 7 Lesson 8.6 Solve the equation b taking the appropriate log of each side. 9 0. 0.. e 6. 5. Use the following propert to solve the equation. For positive numbers b,, and where b, log b log b if and onl if. 5. log log 7 6. log log 9 7. 8. log log 5 ln ln 6 0. Solve the equation b eponentiating each side.. log 5. log 8.. ln 5 5. ln 0 6. e 5 5 8 log log log log log 6 log Compound Interest You deposit $00 in an account that earns % annual interest compounded continuousl. How long does it take the balance to reach the following amounts? 7. $0 8. $50 $00 Algebra 8
8.6 Practice B For use with pages 50 508 Lesson 8.6 Solve the eponential equation. Round the result to three decimal places if necessar.. e 8. 0 50. e. e 8 5. 7 0 6. 5 8 7. 8. e 5 0. e 6 0. e 7. 6.. e 5 5. e 6 6. e 0 7. 6 8. e 8 e 5 0. e 5.. e 5. e. 0. 6 5. e 5 6. e 7. 8 0 Solve the logarithmic equation. Round the result to three decimal places if necessar. 8. ln 5 log 0 0.. 7 ln. log 0 0.. ln 5 5. ln 6. 7. log 0 8. 9 log 0 0. log 5. log.. log 8. log log 5. 6. ln 5 ln 7. ln ln 8. log.5 7 log 0 5 ln 5 log ln 6 8 log log ln 9 ln Compound Interest You deposit $000 into an account that pas % annual interest compounded quarterl. How long will it take for the balance to reach $500? 50. Rocket Velocit Disregarding the force of gravit, the maimum velocit v of a rocket is given b v t ln M, where t is the velocit of the ehaust and M is the ratio of the mass of the rocket with fuel to its mass without fuel. A solid propellant rocket has an ehaust velocit of.5 kilometers per second. Its maimum velocit is 7.5 kilometers per second. Find its mass ratio M. 8 Algebra
8.7 Practice A For use with pages 509 56 Write an eponential function of the form ab whose graph passes through the given points.. 0, ),, 7., 6,,..,,, 8 5., 6, 6, 6 6. Use the table of values to determine whether or not an eponential model is a good fit for the data t,. 7. 8. 0. Solve for.. ln 0.t.60. ln.0t 0.8.. ln.07t. 5. ln.0t 8.9 6. Write a power function of the form a b whose graph passes through the given points. 7.,,, 5 8.,,, Use the table of values to determine whether or not a power function model is a good fit for the data,. 0.. t 5 6 7 8 ln 0. 0.6.07.7.88..7. t 5 6 7 8 ln..5.9.7.88.5. 5.6 t 5 6 7 8 ln 0.05 0.7 0.7 0.0 0.5 0.6 0.75 0.85 t 5 6 7 8 ln..56.8 6.0 7.9 8.9 76.0 ln 0 0.69.099.86.609 ln.6.59.9 5.5 6.58 ln 0 0.69.099.86.609 ln 0.8.9.00.605.05, 0,, 50, 8,, 5 ln.5t 5. ln.0t.6 (,,, 8 Lesson 8.7 Solve for.. ln. ln. ln. ln. ln 0.8 ln Algebra 97
8.7 Practice B For use with pages 509 56 Write an eponential function of the form ab whose graph passes through the given points..,,, 6., 5, 6, 5.,,, 8, 5,, 5. 5. 6.,,, 8, 5 6,, 5 08 Lesson 8.7 Use the table of values to draw a scatter plot of ln versus. Then find an eponential model for the data. 7. 8. 5 6 7 8 8 6 6 8 56 5 0 5 6 7 8.6 8.6 0.76 766 9 86.65 687.97 65. 5 6 7 8.5 6.75 0.5 5.88.78.7 5.58 Write a power function of the form a b whose graph passes through the given points. 0., 6,, 6.,,,., 6,, 86., 6, 9,.., 8, 6, 9,5 5., 879,, 6.070 Use the table of values to draw a scatter plot of ln versus ln. Then find a power model for the data. 6. 7. 5 6 7 8.5 6.5 7.5 5 7.5 96 5 6 7 8. 7.75.99.055.58.9 5.997 66.858 8. Consumer Magazines The table shows the circulation of the top 0 consumer magazines in 997 where represents the magazine s ranking. Use a graphing calculator to find a power model for the data. Use the model to estimate the circulation of the 5th ranked magazine. Rank Circulation Rank Circulation (millions) (millions) 0.5 6 7.65 0. 7 5.05 5.086 8.6.7 9.5 5 0 0.56 98 Algebra
8.8 Practice A For use with pages 57 5 Evaluate the function f for the given value of.. f. f. f 6. 5. 6. f. 7. f 0. 8. f e f 0 f 5 Match the function with its graph. f 0. f e e. A. B. C. f e Identif the horizontal asmptotes of the function. 5. f. f. e e f 6 e Identif the -intercept of the function. 5. 6. 7. e e 5 e Lesson 8.8 Identif the point of maimum growth of the function. 8. f f 0. e e Advertising In Eercises and, use the following information. A compan decides to stop advertising one of its products. The sales of the product S can be modeled b S 00,000 0.5e 0.t where t is the number of ears since advertising stopped.. What are the sales 5 ears after advertising stopped?. What can the compan epect in terms of sales in the future? f e 0 Algebra
8.8 Practice B For use with pages 57 5 Tell whether the function is an eample of eponential growth, eponential deca, logarithmic, or logistics growth.. f. f ln.. f e 5. f.5 6. Match the function with its graph. 7. f 8. f e e A. B. C. f f e f log 6 e Identif the horizontal asmptotes of the function. 0 0. f. f 5. 0.e e Sketch the graph of the function.. f. f 5. e 5e f 0 f e 5 e Solve the equation. 8 6. 7. 8. e 5 e Wildlife Management In Eercises 9, use the following information. A wildlife organization releases 00 deer into a wilderness area. The deer population P can be modeled b P 500 e 0.6t where t is the time in ears. Sketch the graph of the model. 0. Identif the horizontal asmptotes of the graph.. What is the maimum number of deer the wilderness area can support?. What is the deer population after 0 ears? 5e 6 Lesson 8.8 Algebra
CHAPTER 8 SAT/ACT Chapter Test For use after Chapter 8 _. What is the log of 00 to the base 0? A 0 B C D. The natural base e is A rational B imaginar C irrational D undefined. What is the simplified form of A C. What tpe of function is A B C D e 7 e B D Eponential deac function Linear function Quadratic function Eponential growth 5. Which of the following is equivalent to log 5 5? A B 5 C D e e 8e e? f e? 5 8. Which of the following is equivalent to A C Quantitative Comparision Eercises 9 and 0, choose the statement that is true about the given quantities. A The quantit in column A is greater. 0. log b? B C D log b log b log b The quantit in column B is greater. B D The two quantities are equal. log b log b log b log b The relationship cannot be determined from the given information. Column A Column B log 0 000 log 7 Column A The value when 0 of the graph of Column B The value when 0 of the graph of 5 6. What is the solution of the equation 9 7? A No solution B C 5 D 7. What is the asmptote of the graph of f? A -ais B -ais C D Review and Assess Algebra 5
CHAPTER 8 Chapter Review VOCABULARY eponential function, p. 65 base of an eponential function, p. 65 asmptote, p. 65 eponential growth function, p. 66 growth factor, p. 67 eponential deca function, p. 7 deca factor, p. 76 natural base e, or Euler number, p. 80 logarithm of with base b, p. 86 common logarithm, p. 87 natural logarithm, p. 87 change-of-base formula, p. 9 logistic growth function, p. 57 8. EXPONENTIAL GROWTH Eamples on pp. 65 68 EXAMPLE An eponential growth function has the form = ab with a > 0 and b >. To graph = 5 + º, first lightl sketch the graph of = 5, which passes through (0, ) and (, 0). Then translate the graph units to the left and units down. The graph passes through (º, º) and (º, 6). The asmptote is the line = º. The domain is all real numbers, and the range is > º. 5 5 Graph the function. State the domain and range.. = º +. =. = 5 º. = + º 8. EXPONENTIAL DECAY Eamples on pp. 7 76 EXAMPLE An eponential deca function has the form = ab with a > 0 and 0 < b <. To graph =, plot (0, ) and,. From right to left draw a curve that begins just above the -ais, passes through the two points, and moves up. The asmptote is the line = 0. The domain is all real numbers, and the range is > 0. Tell whether the function represents eponential growth or eponential deca. 5. ƒ() = 5 6. ƒ() = 5 7. ƒ() = (6) º 8. ƒ() = () Graph the function. State the domain and range. = 0. = 5 º. = º 5. = º + 7A Chapter 8 Algebra English-Spanish Reviews
8. THE NUMBER e Eamples on pp. 80 8 EXAMPLES You can use e as the base of an eponential function. To graph such a function, use e.78 and plot some points. ƒ() = e is an eponential growth function, since > 0. g() = e º is an eponential deca function, since º < 0. For both functions, the -intercept is, the asmptote is = 0, the domain is all real numbers, and the range is > 0. e e Graph the function. State the domain and range.. = e + 5. = 0.e º 5. = e º 6. = ºe + 8. LOGARITHMIC FUNCTIONS Eamples on pp. 86 89 EXAMPLES You can use the definition of logarithm to evaluate epressions: log b = if and onl if b =. The common logarithm has base 0 (log 0 = log ). The natural logarithm has base e (log e = ln ). To evaluate log 8 096, write log 8 096 = log 8 8 =. To graph the logarithmic function ƒ() = log +, plot points such as (, ) and (0, ). The vertical line = 0 is an asmptote. The domain is > 0, and the range is all real numbers. ƒ( ) log Evaluate the epression without using a calculator. 7. log 6 8. log 8 log 9 0. log 6 Graph the function. State the domain and range.. = log 5. = log. = ln +. = log ( º ) 8.5 PROPERTIES OF LOGARITHMS Eamples on pp. 9 95 EXAMPLES You can use product, quotient, and power properties of logarithms. Epand: log = log º log = log + log º log Condense: log 6 + log 6 = log 6 + log 6 = log 6 (6 ) = log 6 8 Epand the epression. 5. log 6 6. ln 7 7. log 5 8. log 5 Condense the epression. ln º ln 5 0. log + log. 0.5 log + (log 6 º log ) º 8A Chapter 8 Algebra English-Spanish Reviews
8.6 SOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONS Eamples on pp. 50 50 EXAMPLES You can solve eponential equations b equating eponents or b taking the logarithm of each side. You can solve logarithmic equations b eponentiating each side of the equation. 0 =. log = log 0 = log. Take log of each side. log = Eponentiate each side. = log. 0.6 = = 6 Solve the equation. Check for etraneous solutions.. () = 5. e º º = 9. + ln = 8 5. 5 log ( º ) = 8.7 MODELING WITH EXPONENTIAL AND POWER FUNCTIONS Eamples on pp. 509 5 EXAMPLE You can write an eponential function of the form = ab or a power function of the form = a b that passes through two given points. To find a power function given (, ) and (9, ), substitute the coordinates into = a b to get the equations = a b and = a 9 b. Solve the sstem of equations b substitution: a 0. and b.6. So, the function is = 0..6. Find an eponential function of the form = ab whose graph passes through the given points. 6. (, 6), (, 8) 7. (, 8.9), (, 0) 8. (,.), (,.6) Find a power function of the form = a b whose graph passes through the given points. (,.), (6, 7.) 0. (,.5), (,.). (0.5, ), (0, 50) 8.8 LOGISTIC GROWTH FUNCTIONS Eamples on pp. 57 59 EXAMPLE You can graph logistic growth functions b plotting points and identifing important characteristics of the graph. 6 The graph of = is shown. It has asmptotes + e º = 0 and = 6. The -intercept is.5. The point of maimum growth is ln, 6 (0.55, ). (0.55, ) Graph the function. Identif the asmptotes, -intercept, and point of maimum growth.. =. =. = + e º + e º + 0.5e º0.5 9A Chapter 8 Algebra English-Spanish Reviews