About the Portfolio Activities. About the Chapter Project

Transcription

1 Background: Prehistoric rock art from the Canyon de Chelly National Monument, Arizona; Right: Anasazi sandal, years old, found at Navajo National Monument, Arizona About the Chapter Project The heating and cooling of objects can be modeled by functions. Throughout this chapter and in the Chapter Project, Warm Ups,you will model the heating and cooling of a temperature probe over several temperature ranges in order to find an appropriate general model for these phenomena. After completing the Chapter Project, you will be able to do the following: Collect real-world data on the heating and cooling of an object, and determine an appropriate eponential function to model the heating and cooling of an object. Make predictions about the temperature of an object that is heating or cooling to a constant surrounding temperature. Verify Newton s law of cooling. In the Portfolio Activities for Lessons 6. and 6. and in the Chapter Project, you will need to use a program like the one shown on the calculator screen at right to collect temperature data with a CBL. About the Portfolio Activities Throughout the chapter, you will be given opportunities to complete Portfolio Activities that are designed to support your work on the Chapter Project. Using a CBL to collect cooling temperature data in a laboratory setting is included in the Portfolio Activity on page 36. Comparing different models for the cooling temperature data is included in the Portfolio Activity on page 369. Using a CBL to collect warming temperature data and performing appropriate transformations on regression equations are included in the Portfolio Activity on page 38. Comparing Newton s law of cooling with regression models from empirical data is included in the Portfolio Activity on page

2 Eponential Growth and Decay Why Eponential growth and decay can be used to model a number of real-world situations, such as population growth of bacteria and the elimination of medicine from the bloodstream. Objectives Determine the multiplier for eponential growth and decay. Write and evaluate eponential epressions to model growth and decay situations. APPLICATION BIOLOGY Bacteria are very small single-celled organisms that live almost everywhere on Earth. Most bacteria are not harmful to humans, and some are helpful, such as the bacteria in yogurt. Bacteria reproduce, or grow in number, by dividing. The total number of bacteria at a given time is referred to as the population of bacteria. When each bacterium in a population of bacteria divides, the population doubles. Modeling Bacterial Growth TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page 8 You will need: a calculator You can use a calculator to model the growth of 5 bacteria, assuming that the entire population doubles every hour. First enter 5. Then multiply this number by to find the population of bacteria after hour. Repeat this doubling procedure to find the population after hours. 35 CHAPTER 6

3 . Copy and complete the table below. Time (hr) Population CHECKPOINT. Write an algebraic epression that represents the population of bacteria after n hours. (Hint: Factor out 5 from each population figure.) 3. Use your algebraic epression to find the population of bacteria after 0 hours and after 0 hours.. Suppose that the initial population of bacteria was 75 instead of 5. Find the population after 0 hours and after 0 hours. You can represent the growth of an initial population of 00 bacteria that doubles every hour by creating a table Time (hr) Population n () n CONNECTION PATTERNS IN DATA The bar chart at right illustrates how the doubling pattern of growth quickly leads to large numbers Population Time (hr) CHECKPOINT Assuming an initial population of 00 bacteria, predict the population of bacteria after 5 hours and after 6 hours. The population after n hours can be represented by the following eponential epression: n times = 00 n This epression, 00 n,is called an eponential epression because the eponent, n, is a variable and the base,, is a fied number. The base of an eponential epression is commonly referred to as the multiplier. LESSON 6. EXPONENTIAL GROWTH AND DECAY 355

4 Modeling Human Population Growth APPLICATION DEMOGRAPHICS Human populations grow much more slowly than bacterial populations. Bacterial populations that double each hour have a growth rate of 00% per hour. The population of the United States in 990 was growing at a rate of about 8% per decade. In Eample, you will use this growth rate to make predictions. E X A M P L E TECHNOLOGY SCIENTIFIC CALCULATOR The population of the United States was 8,78,30 in 990 and was projected to grow at a rate of about 8% per decade. [Source: U.S. Census Bureau] Predict the population, to the nearest hundred thousand, for the years 00 and 05. SOLUTION. To obtain the multiplier for eponential growth, add the growth rate to 00%. 00% + 8% = 08%, or.08. Write the epression for the population n decades after ,78,30 (.08) n 3. Since the year 00 is decades after 990, substitute for n. 8,78,30(.08) n = 8,78,30(.08) = 90,05,06.3 To the nearest hundred thousand, the predicted population for 00 is 90,00,000. Since the year 05 is 3.5 decades after 990, substitute 3.5 for n. 8,78,30(.08) n = 8,78,30(.08) 3.5 = 35,60,866 To the nearest hundred thousand, the predicted population for 05 is 35,600,000. These predictions are based on the assumption that the growth rate remains a constant 8% per decade. TRY THIS CRITICAL THINKING The population of Brazil was about 6,66,000 in 996 and was projected to grow at a rate of about 7.7% per decade. Predict the population, to the nearest hundred thousand, of Brazil for 06 and 00. [Source: U.S. Census Bureau] If a population s growth rate is % per year,what is the population s growth rate per decade? 356 CHAPTER 6

5 Modeling Biological Decay APPLICATION HEALTH Caffeine is eliminated from the bloodstream of a child at a rate of about 5% per hour. This eponential decrease in caffeine in a child s bloodstream is shown in the bar chart. Original caffeine remaining Caffeine Elimination in Children 00% 80% 60% 0% 0% A rate of decay can be thought of as a negative growth rate. To obtain the multiplier for the decrease in caffeine in the bloodstream of a child, subtract the rate of decay from 00%. Thus, the multiplier is 0.75, as calculated below. 00% 5% = 75%, or Time (hours) E X A M P L E TECHNOLOGY SCIENTIFIC CALCULATOR TRY THIS The rate at which caffeine is eliminated from the bloodstream of an adult is about 5% per hour. An adult drinks a caffeinated soda, and the caffeine in his or her bloodstream reaches a peak level of 30 milligrams. Predict the amount, to the nearest tenth of a milligram, of caffeine remaining hour after the peak level and hours after the peak level. SOLUTION. To obtain the multiplier for eponential decay, subtract the rate of decay from 00%. The multiplier is found as follows: 00% 5% = 85%, or Write the epression for the caffeine level hours after the peak level. 30(0.85) 3. Substitute for. Substitute for. 30(0.85) 30(0.85) = 30(0.85) = 30(0.85) = The amount of caffeine remaining hour after the peak level is 5.5 milligrams. Caffeine is an ingredient in coffee, tea, chocolate, and some soft drinks. The amount of caffeine remaining hours after the peak level is about 5.7 milligrams. A vitamin is eliminated from the bloodstream at a rate of about 0% per hour. The vitamin reaches a peak level in the bloodstream of 300 milligrams. Predict the amount, to the nearest tenth of a milligram, of the vitamin remaining hours after the peak level and 7 hours after the peak level. LESSON 6. EXPONENTIAL GROWTH AND DECAY 357

6 Eercises Communicate. What type of values of n are possible in the bacterial growth epression 5 n and in the United States population growth epression 8,78,30 (.08) n?. Eplain how the United States population growth epression 8,78,30 (.08) n incorporates the growth rate of 8% per decade. 3. What assumption(s) do you make about a population s growth when you make predictions by using an eponential epression?. Describe the difference between the procedures for finding the multiplier for a growth rate of 5% and for a decay rate of 5%. Guided Skills Practice Find the multiplier for each rate of eponential growth or decay. (EXAMPLES AND ) % growth % growth 7. 3% decay % decay Evaluate each epression for = 3. (EXAMPLES AND ) (3) (0.75) APPLICATIONS 3. DEMOGRAPHICS The population of Tokyo-Yokohama, Japan, was about 8,7,000 in 995 and was projected to grow at an annual rate of.%. Predict the population, to the nearest hundred thousand, for the year 00. [Source: U.S. Census Bureau] (EXAMPLE ). HEALTH A certain medication is eliminated from the bloodstream at a rate of about % per hour. The medication reaches a peak level in the bloodstream of 0 milligrams. Predict the amount, to the nearest tenth of a milligram, of the medication remaining hours after the peak level and 3 hours after the peak level. (EXAMPLE ) Practice and Apply Find the multiplier for each rate of eponential growth or decay. 5. 7% growth 6. 9% growth 7. 6% decay 8. % decay % growth 0. 8.% decay. 0.05% decay. 0.08% growth % growth Homework Help Online Go To: go.hrw.com Keyword: MB Homework Help for Eercises 35 Given = 5, y = 3, and z = 3.3, evaluate each epression y () () z 9. 5() y (3) 3. 0() z 3. y () z 3. 00(0.5) 3z (0.5) y 358 CHAPTER 6

7 C H A L L E N G E C O N N E C T I O N Predict the population of bacteria for each situation and time period bacteria that double every hour a. after 3 hours b. after 5 hours bacteria that double every hour a. after 6 hours b. after 8 hours E. coli bacteria that double every 30 minutes a. after hour b. after 6 hours E. coli bacteria that double every 30 minutes a. after hours b. after 3 hours 0. 5 bacteria that triple every hour a. after hour b. after 3 hours. 775 bacteria that triple every hour a. after hours b. after hours. Suppose that you put $500 into a retirement account that grows with an interest rate of 5.5% compounded once each year. After how many years will the balance of the account be at least $5,000? PATTERNS IN DATA Determine whether each table represents a linear, quadratic, or eponential relationship between and y. 0 3 y y y y APPLICATIONS 7. DEMOGRAPHICS The population of Indonesia was 9,56,000 in 990 and was growing at a rate of.9% per year. Predict the population, to the nearest hundred thousand, of Indonesia in 00. [Source: U.S. Census Bureau] 8. HEALTH A dye is injected into the pancreas during a certain medical procedure. A physician injects 0.3 grams of the dye, and a healthy pancreas will secrete % of the dye each minute. Predict the amount of dye remaining, to the nearest hundredth of a gram, in a healthy pancreas 30 minutes after the injection. Bali, Indonesia 9. DEMOGRAPHICS The population of China was,0,005,000 in 996 and was growing at a rate of about 6% per decade. Predict the population, to the nearest hundred thousand, of China in 06 and in 0. [Source: U.S. Census Bureau] LESSON 6. EXPONENTIAL GROWTH AND DECAY 359

8 APPLICATIONS 00% Filter 50. PHYSICAL SCIENCE Suppose that a camera filter transmits 90% of the light striking it, as illustrated at left. a. If a second filter of the same type is added, what portion of light is transmitted through the combination of the two filters? b. Write an epression to model the portion of light that is transmitted through n filters. c. Calculate the portion of light transmitted through, 5, and 6 filters. 90%?%?% Filter Filter 3 5. DEMOGRAPHICS The population of India was 95,08,000 in 996 and was growing at a rate of about.3% per year. [Source: U.S. Census Bureau] a. Predict the population, to the nearest hundred thousand, of India in 000 and in 00. b. Find the growth rate per decade that corresponds to the growth rate of.3% per year. c. Suppose that the population growth rate of India slows to % per year after the year 000. What is the predicted population, to the nearest hundred thousand, of India in 00? 5. CHEMISTRY A dilution is commonly used to obtain the desired concentration of a sample. For eample, suppose that milliliter of hydrochloric acid, or HCl, is combined with 9 milliliters of a buffer. The concentration of the resulting miture is of the 0 original concentration of HCl. a. Suppose that this dilution is performed again with millimeter of the already diluted miture and 9 milliliters of buffer. What is the concentration of the resulting miture (compared with the original concentration)? b. Write an epression to model the concentration of HCl in the resulting miture after repeated dilutions as described in part a. c. What is the concentration of the resulting miture (compared to the original concentration) after 5 repeated dilutions? 53. SPACE SCIENCE The first stage of the Saturn 5 rocket that propelled astronauts to the moon burned about 8% of its remaining fuel every 5 seconds and carried about 600,000 gallons of fuel at liftoff. Estimate the amount of fuel remaining, to the nearest ten thousand gallons, in the first stage minutes after liftoff. 360 CHAPTER 6

9 Look Back Evaluate each epression. (LESSON.) ( ) Simplify each epression, assuming that no variable equals zero. Write your answer with positive eponents only. (LESSON.) m n n 60. a b 3 a b 6. ( y y) 3y Identify each transformation from the graph of f () = to the graph of g. (LESSON.7) 6. g() = g() = ( ) 6. g() = g() = (0.5) g() = ( 3) g() = 5( ) State whether each parabola opens up or down and whether the y-coordinate of the verte is the maimum or minimum value of the function. (LESSON 5.) 68. f() = 69. f() = f() = 3 5 A P P L I C A T I O N Look Beyond 7. INVESTMENTS Suppose that you want to invest $00 in a bank account that earns 5% interest compounded once at the end of each year. Determine the balance after 0 years. PORTFOLIO A C T I T Y I V Refer to the discussions of the Portfolio Activities and Chapter Project on page 353 for background on this activity. You will need a CBL with a temperature probe, a glass of ice water, and a graphics calculator.. First use the CBL to find the temperature of the air. Then place the probe in the ice water for minutes. Record 30 CBL readings taken at -second intervals. Take a final reading at the end of the minutes.. a. Use the linear regression feature on your calculator to find a linear function that models your first 30 readings. (Use the variable t for the time in seconds). b. Use your linear function to predict the temperature of the probe after minutes, or 0 seconds. Compare this prediction with your actual -minute reading. c. Discuss the usefulness of your linear function for modeling the cooling process. (You may want to illustrate your answer with graphs.) Save your data and results for use in the remaining Portfolio Activities. WORKING ON THE CHAPTER PROJECT You should now be able to complete Activity of the Chapter Project. LESSON 6. EXPONENTIAL GROWTH AND DECAY 36

10 Eponential Functions Why You can use eponential functions to calculate the value of investments that earn compound interest and to compare different investments by calculating effective yields. Objectives Classify an eponential function as representing eponential growth or eponential decay. Calculate the growth of investments under various conditions. EXPONENT BASE Consider the function y = and y =.Both functions have a base and an eponent. However, y = is a quadratic function, and y = is an eponential function. In an eponential function, the base is fied and the eponent is variable. Eponential Function The function f() = b is an eponential function with base b,where b is a positive real number other than and is any real number. y y = 3 3 = = = 0 0 = = 8.67 = 3 3 = 8 Eamine the table at left and the graph at right of the eponential function y =. Notice that as -values decrease, the y-values for y = get closer and closer to 0, approaching the -ais as an asymptote. An asymptote is a line that a graph approaches (but does not reach) as its - or y-values become very large or very small. The graph of y = approaches the -ais but never reaches it y = Notice that the domain of y = includes irrational numbers, such as CHAPTER 6

11 Investigating Eponential Functions TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page 8 PROBLEM SOLVING CHECKPOINT You will need: a graphics calculator. Graph y = 3, y =, and y 3 = (.5) on the same screen.. For what value of is y = y = y 3 true? For what values of is y > y > y 3 true? For what values of is y < y < y 3 true? 3. Graph y = ( 3 ), y 5 = ( ), and y 6 = (.5 ) on the same screen as y, y, and y 3.. Look for a pattern.eamine each corresponding pair of functions. y = 3 and y = ( 3) y = and y 5 = ( ) y 3 = (.5) and y 6 = (.5 ) How are the graphs of each corresponding pair of functions related? How are the bases of each corresponding pair of functions related? 5. For what values of b does the graph of y = b rise from left to right? For what values of b does the graph of y = b fall from left to right? The graphs of f() = and g() = ( ) ehibit the two typical behaviors for eponential functions. g() = ( ) is a decreasing eponential function because its base is a positive number less than. g() = ( ) 5 eponential decay 3 y eponential growth f() = f() = is an increasing eponential function because its base is a positive number greater than. y-intercept CONNECTION TRANSFORMATIONS Recall from Lesson.7 that the graphs of f and g are reflections of one another across the y-ais because g() = f( ) = = ( ). Eponential Growth and Decay When b >, the function f() = b represents eponential growth. When 0 < b <, the function f() = b represents eponential decay. LESSON 6. EXPONENTIAL FUNCTIONS 363

12 Eponential growth functions and eponential decay functions of the form y = b have the same domain, range, and y-intercept. For eample: Function Domain Range y-intercept f() = all real all positive numbers real numbers g() = ( ) all real all positive numbers real numbers E X A M P L E Recall from Lesson.7 that y = a f() represents a vertical stretch or compression of the graph of y = f(). This transformation is applied to eponential functions in Eample. Graph f() = along with each function below. Tell whether each function represents eponential growth or eponential decay. Then give the y-intercept. a. y = 3 f() b. y = 5 f( ) TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page 8 CONNECTION TRANSFORMATIONS SOLUTION a. y = 3 f() = 3 The function y = 3 represents eponential growth because the base,, is greater than. The y-intercept is 3 because the graph of f() =,which has a y-intercept of, is stretched by a factor of 3. b. y = 5 f( ) = 5 = 5 ( ) The function y = 5 ( ) represents eponential decay because the base,, is less than. The y-intercept of y = 3 is 3. The y-intercept of y = is. The y-intercept of y = 5 ( ) is 5. The y-intercept is 5 because the graph of f() =,which has a y-intercept of, is stretched by a factor of 5. The y-intercept of y = is. TRY THIS Graph f() = along with each function below. Tell whether each function represents eponential growth or eponential decay. Then give the y-intercept. a. y = 3 f() b. y = f( ) CHECKPOINT What transformation of f occurs when a < 0 in y = a f()? CRITICAL THINKING Describe the effect on the graph of f() = b when b > and b increases. Describe the effect on the graph of f() = b when 0 < b < and b decreases. 36 CHAPTER 6

13 Compound Interest APPLICATION INVESTMENTS The growth in the value of investments earning compound interest is modeled by an eponential function. Compound Interest Formula The total amount of an investment, A, earning compound interest is A(t) = P ( + n r ) nt, where P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years. E X A M P L E Find the final amount of a $00 investment after 0 years at 5% interest compounded annually, quarterly, and daily. SOLUTION In this situation, the principal is $00, the annual interest rate is 5%, and the time period is 0 years. Thus, P = 00, r = 0.05, and t = 0. The table shows calculations for n =, n =, and n = 365. TECHNOLOGY SCIENTIFIC CALCULATOR Compounding n period A(0) = 00 ( + 0ṅ 05 ) n 0 Final amount annually A(0) = 00 ( ) 0 $6.89 quarterly A(0) = 00 ( ) 0 $6.36 daily 365 A(0) = 00 ( ) $6.87 CHECKPOINT Describe what happens to the final amount as the number of compounding periods increases. Effective Yield APPLICATION INVESTMENTS Suppose that you buy an item for $00 and sell the item one year later for $05. In this case, the effective yield of your investment is 5%. The effective yield is the annually compounded interest rate that yields the final amount of an investment. You can determine the effective yield by fitting an eponential regression equation to two points. LESSON 6. EXPONENTIAL FUNCTIONS 365

14 E X A M P L E 3 CONNECTION STATISTICS A collector buys a painting for $00,000 at the beginning of 995 and sells it for $50,000 at the beginning of 000. Use an eponential regression equation to find the effective yield. TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page 9 SOLUTION. Find the eponential equation that represents this situation. To find effective yield, the interest is compounded annually, so n =. From 995 to 000 is 5 years, so t = 5. A(t) = P ( + n r ) nt 50,000 = 00,000 ( + r ) 5 50,000 = 00,000( + r) 5. Enter the two points that represent the given information, (0, 00,000) and (5, 50,000). Find and graph the eponential regression equation that fits the points. 3. The multiplier is about.08, so the effective yield, is about.08 = 0.08, or 8.%. (0, 00,000) The eponential regression equation is y 00,000(.08). (5, 50,000) TRY THIS Find the effective yield for a painting bought for $00,000 at the end of 99 and sold for $00,000 at the end of 00. Eercises Communicate Activities Online Go To: go.hrw.com Keyword: MB Medicine. If b > 0 and the graph of y = b falls from left to right, describe the possible values of b.. Compare the domain and range of y = 3 with the domain and range of y = ( 3 ). 3. Describe how the y-intercept of the graph of f() = (5) is related to the value of a in f() = ab.. How are the functions y = and y = similar, and how are they different? 366 CHAPTER 6

15 Guided Skills Practice Tell whether each function represents eponential growth or eponential decay, and give the y-intercept. (EXAMPLE ) 5. f() = ( ) 6. g() = 3() 7. k() = 5(0.5) APPLICATIONS 8. INVESTMENTS Find the final amount of a $50 investment after 5 years at 6% interest compounded annually, quarterly, and daily. (EXAMPLE ) 9. INVESTMENTS Find the effective yield for a $000 investment that is worth $000 after 5 years. (EXAMPLE 3) Practice and Apply Identify each function as linear, quadratic, or eponential. 0. g() = k() = (77 ).. f() = (.5) 3. k() = g() = (00) h() = Tell whether each function represents eponential growth or decay. 6. y() = (.5) 7. k() = 500(.5) 8. y(t) = 5 ( ) t 9. d() = 0.5 ( ) 0. g() = 0.5(0.8). s(k) = 0.5(0.5) k. m() = (0.9) 3. f(k) = 7 k. g() = 0.5(787) Match each function with its graph. 5. y = 6. y = (3) 7. y = ( 3 ) 8. y = ( ) y a b c d Find the final amount for each investment. 9. $000 at 6% interest compounded annually for 0 years 30. $000 at 6% interest compounded semiannually for 0 years 3. $750 at 0% interest compounded quarterly for 0 years 3. $750 at 5% interest compounded quarterly for 0 years 33. $800 at 5.65% interest compounded daily for 3 years 3. $800 at 5.65% interest compounded daily for 6 years 35. Graph f() =, g() = 5, and h() = 8. a. Which function ehibits the fastest growth? the slowest growth? b. What is the y-intercept of each function? c. State the domain and range of each function. 36. Graph a() = ( ), b() = ( 5 ), and c() = ( 8 ). a. Which function ehibits the fastest decay? the slowest decay? b. What is the y-intercept of each function? c. State the domain and range of each function. C H A L L E N G E 37. Describe when the graph of f() = ab is a horizontal line. LESSON 6. EXPONENTIAL FUNCTIONS 367

16 CONNECTIONS Homework Help Online Go To: go.hrw.com Keyword: MB Homework Help for Eercises 38 6 TRANSFORMATIONS Graph each pair of functions and describe the transformations from f to g. 38. f() = ( ) and g() = 5 ( ) 39. f() = ( 0 ) and g() = 0.5 ( 0 ) 0. f() = and g() = 3() +. f() = 0 and g() = (0) 3. f() = 0 and g() = 3(0) + 3. f() = and g() = 5(). f() = 3 ( ) and g() = 3( ) 5. f() = ( 3 ) and g() = (3) 6. TRANSFORMATIONS Describe how each transformation of f() = b affects the domain and range, the asymptotes, and the intercepts. a. a vertical stretch b. a vertical compression c. a horizontal translation d. a vertical translation e. a reflection across the y-ais STATISTICS Use an eponential regression equation to find the effective yield for each investment. Assume that interest is compounded only once each year. 7 a $000 mutual fund investment made at the beginning of 990 that is worth $50 at the beginning of a house that is bought for $75,000 at the end of 995 and that is worth $95,000 at the end of 005 STATISTICS Use an eponential regression equation to model the annual rate of inflation, or percent increase in price, for each item described. 9 a half-gallon of milk cost $.37 in 989 and $.8 in 995 [Source: U.S. Bureau of Labor Statistics] 50 a gallon of regular unleaded gasoline cost $0.93 in 986 and $. in 993 [Source: U.S. Bureau of Labor Statistics] APPLICATIONS 5. INVESTMENTS Find the final amount of a $000 certificate of deposit (CD) after 5 years at an annual interest rate of 5.5% compounded annually. Certificate of Deposit 5. INVESTMENTS Consider a $000 investment that is compounded annually at three different interest rates: 5%, 5.5%, and 6%. a. Write and graph a function for each interest rate over a time period from 0 to 60 years. b. Compare the graphs of the three functions. c. Compare the shapes of the graphs for the first 0 years with the shapes of the graphs between 50 and 60 years. 53. INVESTMENTS The final amount for $5000 invested for 5 years at 0% annual interest compounded semiannually is $57,337. a. What is the effect of doubling the amount invested? b. What is the effect of doubling the annual interest rate? c. What is the effect of doubling the investment period? d. Which of the above has the greatest effect on the final amount of the investment? 368 CHAPTER 6

17 Look Back Find the inverse of each function. State whether the inverse is a function. (LESSON.5) 5. {(, ), ( 3, ), (, ), (3, )} 55. {(7, ), (3, ), (, ), (0, 0)} 56. y = ( + 3) 57. y = y = y = Graph each piecewise function. (LESSON.6) 60. f() = 9 if 0 < 5 6. g() if 5 < 0 = 3 Let A =, B =, and C = product matri, if it eists. (LESSON.) 6. AB 63. BA 6. AC 65. CA 66. BC 67. CB if 0 < if < 5 if 5 < 0. Find each Find a quadratic function to fit each set of points eactly. (LESSON 5.7) 68. (, ), (, 5), (3, 3) 69. (0, ), (, 5), (3, 5) Look Beyond 70. Use guess-and-check to find such that 0 = 50. PORTFOLIO A C T I T Y I V For this activity, use the data collected in the Portfolio Activity on page 36.. a. Use the quadratic regression feature on your calculator to find a quadratic function that models your first 30 readings. b. Use your quadratic function to predict the temperature of the probe after minutes. Compare this prediction with your actual -minute reading. c. Discuss the usefulness of your quadratic function for modeling the cooling process.. Now use the eponential regression feature on your calculator to find an eponential function that models your first 30 readings, and repeat parts b and c of Step. Save your data and results to use in the remaining Portfolio Activities. WORKING ON THE CHAPTER PROJECT You should now be able to complete Activity of the Chapter Project. LESSON 6. EXPONENTIAL FUNCTIONS 369

18 Logarithmic Functions Objectives Write equivalent forms for eponential and logarithmic equations. Use the definitions of eponential and logarithmic functions to solve equations. Why Logarithmic functions are widely used in measurement scales such as the ph scale, which ranges from 0 to. Substance ph gastric fluid.8 lemon juice.. vinegar. 3. banana.8 saliva water 7 egg white Rolaids, Tums 9.9 milk of magnesia 0.5 The ph of an acidic solution is less than 7, the ph of a basic solution is greater than 7, and the ph of a neutral solution is 7. Logarithms are used to find unknown eponents in eponential models. Logarithmic functions define many measurement scales in the sciences, including the ph, decibel, and Richter scales. Approimating Eponents TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page 9 CHECKPOINT PROBLEM SOLVING CHECKPOINT You will need: a graphics calculator Use the table below to complete this Activity y = How are the -values in the table related to the y-values?. Use the table above to find the value of in each equation below. a. 0 = 000 b. 0 = 00 c. 0 = 0 00 d. 0 = 3. Make a table of values for y = 0.Use the table to approimate the solution to 0 = 7 to the nearest hundredth.. Use a table of values to approimate the solution to 0 = 85 to the nearest hundredth. 370 CHAPTER 6

19 A table of values for y = 0 can be used to solve equations such as 0 = 000 and 0 = 00. However, to solve equations such as 0 = 85 or 0 =.3, a logarithm is needed. With logarithms, you can write an eponential equation in an equivalent logarithmic form. Eponential form Logarithmic form 0 3 = = log eponent base Equivalent Eponential and Logarithmic Forms For any positive base b,where b : b = y if and only if = log b y E X A M P L E a. Write 5 3 = 5 in logarithmic form. b. Write log 3 8 = in eponential form. SOLUTION a. 5 3 = 5 3 = log is the eponent and 5 is the base. b. log 3 8 = 3 = 8 3 is the base and is the eponent. TRY THIS Copy and complete each column in the table below. Eponential form Logarithmic form 5 = 3? 3 = 9?? log = 3? log 6 = E X A M P L E TECHNOLOGY SCIENTIFIC CALCULATOR TRY THIS You can evaluate logarithms with a base of 0 by using the calculator. key on a Solve for 0 = 85 for.round your answer to the nearest thousandth. SOLUTION Write 0 = 85 in logarithmic form, and use the Because 0 = 0 and 0 = 00,.99 is a reasonable answer. 0 = 85 = log key. Use a calculator. Solve 0 = for.round your answer to the nearest thousandth. 09 LOG LOG LESSON 6.3 LOGARITHMIC FUNCTIONS 37

20 Definition of Logarithmic Function The inverse of the eponential function y = 0 is = 0 y.to rewrite = 0 y in terms of y, use the equivalent logarithmic form, y = log 0. Eamine the tables and graphs below to see the inverse relationship between y = 0 and y = log 0. y = y = log y y = 0 (, 0) y = y = log 0 (0, ) The table below summarizes the relationship between the domain and range of y = 0 and of y = log 0. Function Domain Range y = 0 all real numbers all positive real numbers y = log 0 all positive real numbers all real numbers Logarithmic Functions The logarithmic function y = log b with base b,or = b y,is the inverse of the eponential function y = b,where b and b > 0. CRITICAL THINKING Describe the graph that results if b = in y = log b.is y = log a function? Because y = log b is the inverse of the eponential function y = b and y = log b is a function, the eponential function y = b is a one-to-one function. This means that for each element in the domain of an eponential function, there is eactly one corresponding element in the range. For eample, if 3 = 3,then =. This is called the One-to-One Property of Eponents. One-to-One Property of Eponents If b = b y,then = y. 37 CHAPTER 6

21 E X A M P L E 3 Find the value of v in each equation. a. v = log 5 5 b. 5 = log v 3 c. = log 3 v SOLUTION Write the equivalent eponential form, and solve for v. a. v = log 5 5 b. 5 = log v 3 c. = log 3 v 5 v = 5 v 5 = 3 3 = v (5 3 ) v = 5 v 5 = 5 8 = v 5 3v = 5 v = 3v = v = 3 Apply the One-to-One Property. TRY THIS Find the value of v in each equation. a. v = log 6 b. = log v 5 c. 6 = log 3 v CONNECTION TRANSFORMATIONS Recall from Lesson.7 that the graph of y = f() is the graph of y = f() reflected across the -ais. The graph of y = log 0 and of its reflection across the -ais, y = log 0,are shown at right. y y = log y = log 0 The function y = log 0 is used in chemistry to measure ph levels. The ph of a solution describes its acidity. Substances that are more acidic have a lower ph, while substances that are less acidic, or basic, have a higher ph. The ph of a substance is defined as ph = log 0 [H + ], where [H + ] is the hydrogen ion concentration of a solution in moles per liter. E X A M P L E APPLICATION CHEMISTRY TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page 9 TRY THIS The ph of a carbonated soda is 3. What is [H + ] for this soda? SOLUTION ph = log 0 [H + ] 3 = log 0 [H + ] Substitute 3 for ph. 3 = log 0 [H + ] 0 3 = [H + ] Write the equivalent eponential equation. CHECK Graph y = log 0 and y = 3 on the same screen, and find the point of intersection. The window at right shows -values between 0 and 0.0. Thus, there is 0,or 0.00, moles of hydrogen 00 ions in a liter of carbonated soda that has a ph of 3. Find [H + ] for orange juice that has a ph of LESSON 6.3 LOGARITHMIC FUNCTIONS 373

22 Eercises Communicate. Describe the relationship between logarithmic functions and eponential functions.. State the domain and range of logarithmic functions. How are they related to the domain and range of eponential functions? 3. Eplain how to approimate the value of in = 58 by using the table feature of a graphics calculator. Guided Skills Practice. Write = 6 in logarithmic form. (EXAMPLE ) 5. Write log 5 5 = in eponential form. (EXAMPLE ) Solve each equation for. Round your answers to the nearest thousandth. (EXAMPLE ) 6. 0 = = 5 00 A P P L I C A T I O N Find the value of v in each equation. (EXAMPLE 3) 8. v = log = log v 0. = log v. CHEMISTRY The ph of black coffee is 5. What is [H + ] for this coffee? (EXAMPLE ) Practice and Apply Write each equation in logarithmic form.. = 3. 5 = = = = = = = 9 0. ( ) 3 = 6. ( 9 ) = 8. ( 3 ) = 9 3. ( ) 3 = 8 Write each equation in eponential form.. log 6 36 = 5. log = 3 6. log = 3 7. log 0 0. = 8. 3 = log = log log 3 8 = = log log 3 = 5 = 3. = log = Find the approimate value of each logarithmic epression. 36. log log log log 0, log log log log CHAPTER 6

23 Solve each equation for. Round your answers to the nearest hundredth.. 0 = = 6. 0 = = = = = = = = = = Find the value of v in each equation. 56. v = log v = log v = log v = log v = log v = log v = log v = log v = log 65. v = log v = log 67. v = log = log 6 v 69. = log 7 v 70. = log 5 v 7. = log 3 v 7. = log 9 v = log 8 v 7. = log 6 v = log v = log 3 v = log v 78. log v 6 = 79. log v 5 = log v 9 = 8. log v = 3 8. log v = log v 8 = 3 8. log v 6 = log v 3 = Graph f() = 3 along with f.make a table of values that illustrates the relationship between f and f. 87. Graph f() = 3 along with f.make a table of values that illustrates the relationship between f and f. CHALLENGES Homework Help Online Go To: go.hrw.com Keyword: MB Homework Help for Eercises 9 96 ph paper turns red in an acidic solution, 0 < ph < 7; the paper turns green in a neutral solution, indicating a ph of 7; and the paper turns blue in a basic solution, 7 < ph <. Find the value of each epression. 88. log log log 8 TRANSFORMATIONS Let f() = log 0. For each function, identify the transformations from f to g. 9. g() = 3 log 0 9. g() = 5 log g() = log g() = 0.5 log g() = log 0 ( ) 96. g() = log 0 ( + 5) 3 CHEMISTRY Calculate [H + ] for each of the following: 97. household ammonia with a ph of about distilled water with a ph of human blood with a ph of about CHEMISTRY How much greater is [H + ] for lemon juice, which has a ph of., than [H + ] for water, which has a ph of 7.0? Battery acid Stomach acid, lemons Tomatoes, bananas Black coffee Pure water Baking soda Hand soap Household ammonia Lye (drain cleaner) LESSON 6.3 LOGARITHMIC FUNCTIONS 375

24 A P P L I C A T I O N 0. PHYSICS Earth s atmosphere is like an ocean of air with the upper layers of air pressing down on the lower layers of air. The weight of the layers of air creates atmospheric air pressure. At sea level (altitude of zero), the average air pressure is about.7 pounds per square inch. The air pressure, P,decreases with altitude, a,in feet according to the function P =.7(0) a.Find the altitude that corresponds to the air pressure commonly found in commercial airplanes,.8 pounds per square inch. Look Back 0. Write a linear equation for a line with a slope of and a y-intercept of 3. (LESSON.) State the property that is illustrated in each statement. All variables represent real numbers. (LESSON.) 03. (5y) = 5y 0. ( + z) + y = + (z + y) 05. (3) = 3() = a 07. =, where a 0 a = + ( 3) 09 Find the inverse of the matri 3. (LESSON.3) 0. Solve the quadratic equation = 0. (LESSONS 5. AND 5.). State the two solutions of the equation + = 0. (LESSON 5.6). If an interest rate is 7.3%, what is the multiplier? (LESSON 6.) Look Beyond 3. Calculate log + log 8 and log 3 log. Then compare these values with the value of log CHAPTER 6

25 Properties of Logarithmic Functions Why The properties of logarithms allow you to simplify logarithmic epressions, which makes evaluating the epressions easier. Objectives Simplify and evaluate epressions involving logarithms. Solve equations involving logarithms. John Napier (550 67) Title page and calculations from Napier s Mirifici Logarithmorum Canonis Descriptio In the seventeenth century, a Scottish mathematician named John Napier developed methods for efficiently performing calculations with large numbers. He found a method for finding the product of two numbers by adding two corresponding numbers, which he called logarithms. John Napier s contributions to mathematics are contained in two essays: Mirifici Logarithmorum Canonis Descriptio (Description of the Marvelous Canon of Logarithms), published in 6, and Mirifici Logarithmorum Canonis Constructio (Construction of the Marvelous Canon of Logarithms), published in 69, two years after his death. LESSON 6. PROPERTIES OF LOGARITHMIC FUNCTIONS 377

26 Product and Quotient Properties of Logarithms The Product, Quotient, and Power Properties of Eponents are as follows: a m a n = a m + n Product Property a m an = a m n Quotient Property (a m ) n = a m n Power Property Each property of eponents has a corresponding property of logarithms. Eploring Properties of Logarithms You will need: no special tools Use the following table to complete the activity: y = log CHECKPOINT CHECKPOINT. The epression log ( ) can be written as log 8. Use this fact and the table above to evaluate each epression below. a. log ( ) = n?n and log + log = n?n b. log ( 8) = n?n and log + log 8 = n?n c. log ( 6) = n?n and log + log 6 = n?n d. log ( 3) = n?n and log + log 3 = n?n. In Step, how is the first epression in each pair related to the second epression? Use this pattern to make a conjecture about log (a b). 3. The epression log 6 can be written as log 8. Use this fact and the table above to evaluate each epression below. a. log = 6 n?n and log 6 log = n?n b. log 6 3 = n?n and log 6 log 3 = n?n c. log 3 = 8 n?n and log 3 log 8 = n?n d. log 8 = n?n and log 8 log = n?n. In Step 3, how is the first epression in each pair related to the second epression? Use this pattern to make a conjecture about log a b. The patterns eplored in the Activity illustrate the Product and Quotient Properties of Logarithms given below. Product and Quotient Properties of Logarithms For m > 0, n > 0, b > 0, and b : Product Property log b (mn) = log b m + log b n Quotient Property log b m n = log b m log b n 378 CHAPTER 6

27 E X A M P L E You can use the Product and Quotient Properties of Logarithms to evaluate logarithmic epressions. This is shown in Eample. Given log , approimate the value of each epression below by using the Product and Quotient Properties of Logarithms. a. log b. log.5 SOLUTION a. log = log ( 3) b. log.5 = log 3 = log + log + log 3 = log 3 log TRY THIS Given that log 3 =.5850, approimate each epression below by using the Product and Quotient Properties of Logarithms. a. log 8 b. log 3 E X A M P L E Eample demonstrates how to use the properties of logarithms to rewrite a logarithmic epression as a single logarithm. Write each epression as a single logarithm. Then simplify, if possible. a. log 3 0 log 3 5 b. log b u + log b v log b uw SOLUTION a. log 3 0 log 3 5 = log 3 0 b. log 5 b u + log b v log b uw = log b uv log b uw = log 3 = log uv b uw = log v b w TRY THIS Write each epression as a single logarithm. Then simplify if possible. a. log 8 log 6 b. log b log b 3y + log b y The Power Property of Logarithms Eamine the process of rewriting the epression log b (a ). log b (a ) = log b (a a a a) = log b a + log b a + log b a + log b a = log b a This illustrates the Power Property of Logarithms given below. Power Property of Logarithms For m > 0, b > 0, b, and any real number p: log b m p = p log b m LESSON 6. PROPERTIES OF LOGARITHMIC FUNCTIONS 379

28 E X A M P L E 3 In Eample 3, the Power Property of Logarithms is used to simplify powers. Evaluate log 5 5. SOLUTION log 5 5 = log 5 5 = = 8 Use the Power Property of Logarithms. TRY THIS Evaluate log Eponential-Logarithmic Inverse Properties Recall from Lesson.5 that functions f and g are inverse functions if and only if (f g)() = and (g f )() =.The functions f() = log b and g() = b are inverses, so (f g)() = log b b = and (g f )() = b log b =. Eponential-Logarithmic Inverse Properties For b > 0 and b : log b b = and b log b = for > 0 E X A M P L E Evaluate each epression. a. 3 log 3 + log 5 5 b. log 3 5 log 5 3 SOLUTION a. 3 log 3 +log 5 5 b. log 3 5 log 5 3 = + log 5 5 = log 5 3 = + = 5 3 = 6 = TRY THIS CRITICAL THINKING Evaluate each epression. a. 7 log 7 log 3 8 b. log log 3 8 Verify the Eponential-Logarithmic Inverse Properties by using only the equivalent eponential and logarithmic forms given on page 37. Because eponential functions and logarithmic functions are one-to-one functions, for each element in the domain of y = log,there is eactly one corresponding element in the range of y = log. One-to-One Property of Logarithms If log b = log b y,then = y. 380 CHAPTER 6

29 E X A M P L E 5 Solve log 3 ( + 7 5) = log 3 (6 + ) for.check your answers. SOLUTION log 3 ( + 7 5) = log 3 (6 + ) = 6 + Use the One-to-One Property of Logarithms. + 6 = 0 ( )( + 3) = 0 = or = 3 Use the Zero Product Property. CHECK Let =. Let = 3. log 3 ( + 7 5) =? log 3 (6 + ) log 3 ( + 7 5) =? log 3 (6 + ) log 3 3 = log 3 3 log 3 ( 7) = log 3 ( 7) True Undefined Since the domain of a logarithmic function ecludes negative numbers, the solution cannot be 3. Therefore, the solution is. Eercises Communicate. Given that log , eplain how to approimate the values of log and log Eplain how to write an epression such as log 7 3 log 7 as a single logarithm. 3. Eplain how to evaluate log 8 and log 7.Include the names of the properties you would use.. Eplain why you must check your answers when solving an equation such as log 3 = log ( + ) for. Guided Skills Practice Given log , approimate the value for each logarithm by using the Product and Quotient Properties of Logarithms. (EXAMPLE ) 5. log log Write each epression as a single logarithm. Then simplify, if possible. (EXAMPLE ) 7. log 3 log 3 y + log 3 z 8. log 3 + log 6 log 0 Evaluate each epression. (EXAMPLES 3 AND ) 9. log log 3. log Solve log 3 = log 3 ( ) for, and check your answers. (EXAMPLE 5) LESSON 6. PROPERTIES OF LOGARITHMIC FUNCTIONS 38

30 Homework Help Online Go To: go.hrw.com Keyword: MB Homework Help for Eercises 3 6, 9 Practice and Apply Write each epression as a sum or difference of logarithms. Then simplify, if possible. 3. log 8 (5 8). log 8y 5. log log 3 Use the values given below to approimate the value of each logarithmic epression in Eercises 7 8. log log 5.39 log 5.60 log log log log 5 8. log log 8 0. log. log 60. log log log log log log 5 8. log 7 Write each epression as a single logarithm. Then simplify, if possible. 9. log 5 + log log 8 + log 3. log 3 5 log log log log 5 + log log 0 3. log 3 + log 3 log log 7 3 log log 7 6y 36. log 5 6s log 5 s + log 5 t log m log n log 3 y log log b m + log b n 3 log b p 0. log b 3c + log b d log b 5e. log 7. + log 3 Evaluate each epression log log 9 5. log 5 6. log log log log log log 9 9 log log log log 6 3 log log 3 + log log 3 9 log 3 5. log 8 log 7 Solve for, and check your answers. Justify each step in the solution process. 55. log 7 = log ( + ) 56. log 5 (3 ) = log log b ( 5) = log b (6 + ) 58. log 0 (5 3) log 0 ( + ) = log a + log a = log a (5 + 3) 60. log b ( ) + log b 6 = log b 6 6. log 3 + log 3 5 = log 3 ( + 3) 6. log 5 + log 5 t = log 5 (3 t) State whether each equation is always true, sometimes true, or never true. Assume that is a positive real number. 63. log 3 9 = log log 8 log = 65. log = log 66. log log 5 = log log 3 log 67. = 5 log 3 log 68. log( ) = log l og 69. log = log 70. log = log 7. log 3 + log 3 = log 3 38 CHAPTER 6

31 C H A L L E N G E APPLICATIONS Solve each equation. 7. log (log 3 ) = log 6 [log 5 (log 3 )] = 0 7. HEALTH The surface area of a person is commonly used to calculate dosages of medicines. The surface area of a child is often calculated with the following formula, where S is the surface area in square centimeters, W is the child s weight in kilograms, and H is the child s height in centimeters. log 0 S = 0.5 log 0 W log 0 H + log Use the properties of logarithms to write a formula for S without logarithms. 75. PHYSICS Atmospheric air pressure, P,in pounds per square inch and altitude, a,in feet are related by the logarithmic equation a = 55, log P 0.Use properties of logarithms to find how much.7 greater the air pressure at the top of Mount Whitney in the United States is compared with the air pressure at the Mount Everest top of Mount Everest on the border of 9,08 feet Tibet and Nepal. The altitude of Mount Everest is 9,08 feet, and the altitude of Mount Whitney is,95 feet. (Hint: Find the ratio of the air pressures.) Mount Whitney, 95 feet Sea level Look Back Minimize each objective function under the given constraints. (LESSON 3.5) 76. Objective function: C = + 5y 77. Objective function: C = + y + 5y 8 y y 3 7y Constraints: 0 Constraints: 0 y 0 y 0 Write the matri equation that represents each system. (LESSON.) 3 + y z = y + z = 8 y = y z = 3 + y + y + 3 z = 6 = 3 3y z + = 5 y + 3z = 33 LESSON 6. PROPERTIES OF LOGARITHMIC FUNCTIONS 383

32 APPLICATIONS Write the augmented matri for each system of equations. (LESSON.5) 3 6y + 3z = y = y = y = z 8.5y +.z =. = 5 y y + z = 5 3.3z +.3 = 9 8. INVESTMENTS An investment of $00 earns an annual interest rate of 5%. Find the amount after 0 years if the interest is compounded annually, quarterly, and daily. (LESSON 6.) 85. INVESTMENTS Find the final amount after 8 years of a $500 investment that is compounded semiannually at 6%, 7%, and 8% annual interest. (LESSON 6.) Look Beyond 86 e is an irrational number between and 3. The epression log e is commonly written as ln.use the LN key to solve e 3 = 5 for to the nearest hundredth. PORTFOLIO A C T I T Y I V In this activity, you will use the CBL to collect data as a warm probe cools in air.. First record the air temperature for reference. Then place the temperature probe in hot water until it reaches a reading of at least 60 C. Remove the probe from the water and record the readings as in Step of the Portfolio Activity on page 36.. a. Using the regression feature on your calculator, find linear, quadratic, and eponential functions that model this data. (Use the variable t for time in seconds.) b. Use each function to predict the temperature of the probe after minutes (0 seconds). Compare the predictions with the actual -minute reading. c. Discuss the usefulness of each function for modeling the cooling process. 3. Create a new function to model the cooling process by performing the steps below. a. Subtract the air temperature from each temperature recorded in your data list. Store the resulting data values in a new list. b. Use the eponential regression feature on your calculator to find an eponential function of the form y = a b t that models this new data set. c. Add the air temperature to the function you found in part b.graph the resulting function, y = a b t + c,which will be called the approimating function. d. Repeat parts b and c from Step with the approimating function. Save your data and results to use in the last Portfolio Activity. WORKING ON THE CHAPTER PROJECT You should now be able to complete Activity 3 of the Chapter Project. 38 CHAPTER 6

33 The Activity below leads to a method for evaluating logarithmic epressions with bases other than 0. Eploring Change of Base CHECKPOINT You will need: a scientific calculator. Write 3 = 8 as a logarithmic epression for in base 3.. Write 3 = 8 as a logarithmic epression for in base 0. (Hint: Refer to Eample 3.) 3. Set your epressions for from Steps and equal to each other.. Write b = y in logarithmic form. Then solve b = y for, and give the result as a quotient of logarithms. Set your two resulting epressions equal to each other. The answer to Step in the Activity suggests a change-of-base formula, shown below, for writing equivalent logarithmic epressions with different bases. Change-of-Base Formula For any positive real numbers a, b, and > 0: oga log b = l log b a CHECKPOINT Write log 9 7 as a base 3 epression. E X A M P L E TECHNOLOGY SCIENTIFIC CALCULATOR You can use the change-of-base formula to change a logarithmic epression of any base to base 0 so that you can use the LOG key on a calculator. This is shown in Eample. Evaluate log Round your answer to the nearest hundredth. SOLUTION Use the change-of-base formula to change from base 7 to base 0. log 7 56 = l og 56 log Use a calculator to evaluate. TRY THIS CRITICAL THINKING Evaluate log Round your answer to the nearest hundredth. Use the change-of-base formula to justify each formula below. a. (log a b)(log b c) = log a c b. log a b = log b a 388 CHAPTER 6

34 Eercises Communicate. Eplain why a common logarithmic function is appropriate to use for the decibel scale of sound intensities.. Describe the steps you would take to solve 6 = 39 for. 3. Eplain how to evaluate log 9 by using a calculator. APPLICATIONS Guided Skills Practice. PHYSICS Suppose that a soft whisper is about 75 times as loud as the threshold of hearing, I 0.Find the relative intensity, R,ofthis whisper in decibels. (EXAMPLE ) 5. PHYSICS The relative intensity, R,ofa loud siren is about 30 decibels. Compare the intensity of this siren with the threshold of hearing, I 0. (EXAMPLE ) Solve each eponential equation for. Round your answers to the nearest hundredth. (EXAMPLE 3) 6. 8 = 7. = 7 Evaluate each logarithmic epression. Round your answers to the nearest hundredth. (EXAMPLE ) 8. log 6 9. log 5 Practice and Apply Homework Help Online Go To: go.hrw.com Keyword: MB Homework Help for Eercises 0 7 Solve each equation. Round your answers to the nearest hundredth. 0. = 7. = 9. 7 = = = = = = = = = =.9. + = = 8. 6 = = = = 360 LESSON 6.5 APPLICATIONS OF COMMON LOGARITHMS 389

35 C H A L L E N G E APPLICATIONS Evaluate each logarithmic epression to the nearest hundredth. 8. log 9 9. log log log log log 5 3. log log log log log log 8 0. log 8. log log 6 3. log log log Prove that log (bn ) = n logb is true. 7. PHYSICS The sound of a leaf blower is about times the intensity of the threshold of hearing, I 0.Find the relative intensity, R,ofthis leaf blower in decibels. 8. PHYSICS The sound of a conversation is about 350,000 times the intensity of the threshold of hearing, I 0.Find the relative intensity, R,ofthis conversation in decibels. Activities Online Go To: go.hrw.com Keyword: MB Stock Closings 9. PHYSICS Suppose that the relative intensity, R,ofa rock band is about 5 decibels. Compare the intensity of this band with that of the threshold of hearing, I PHYSICS The relative intensity, R,ofan automobile engine is about 55 decibels. Compare the intensity of this engine with that of the threshold of hearing, I PHYSICS Suppose that background music is adjusted to an intensity that is 000 times as loud as the threshold of hearing. What is the relative intensity of the music in decibels? 5. PHYSICS Suppose that a burglar alarm has a rating of 0 decibels. Compare the intensity of this decibel rating with that of the threshold of hearing, I PHYSICS Simon Robinson set the world record for the loudest scream by producing a scream of 8 decibels at a distance of 8 feet and inches. Compare the intensity of this decibel rating with that of the threshold of hearing, I 0.[Source: The Guinness Book of World Records, 997] 5. PHYSICS A small jet engine produces a sound whose intensity is one billion times as loud as the threshold of hearing. What is the relative intensity of the engine s sound in decibels? 390 CHAPTER 6

36 The Natural Base, e The model below shows the embryo inside an 8-inch dinosaur egg, the largest known. Objectives Evaluate natural eponential and natural logarithmic functions. Model eponential growth and decay processes. Why The eponential function with base e and its inverse, the natural logarithmic function, have a wide variety of real-world applications. For eample, these functions are used to estimate the ages of artifacts found at archaeological digs. The natural base, e, is used to estimate the ages of artifacts and to calculate interest that is compounded continuously. Recall from Lesson 6. the compound interest formula, A(t) = P ( + n r ) nt,where P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is the time in years. This formula is used in the Activity below. Investigating the Growth of $ APPLICATION INVESTMENTS You will need: a scientific calculator. Copy and complete the table below to investigate the growth of a $ investment that earns 00% annual interest (r = ) over year (t = ) as the number of compounding periods per year, n,increases. Use a calculator, and record the value of A to five places after the decimal point. Compounding schedule n ( + n ) n Value, A annually ( + ) semiannually ( + ) quarterly monthly daily 365 hourly every minute every second CHECKPOINT. Describe the behavior of the sequence of numbers in the Value column. As n becomes very large, the value of ( + n ) n approaches the number ,named e.because e is an irrational number like π, its decimal epansion continues forever without repeating patterns. 39 CHAPTER 6

37 The Natural Eponential Function CHECKPOINT The eponential function with base e, f() = e, is called the natural eponential function and e is called the natural base. The function f() = e is graphed at right. Notice that the domain is all real numbers and the range is all positive real numbers. What is the y-intercept of the graph of f() = e? Natural eponential functions model a variety of situations in which a quantity grows or decays continuously. Eamples that you will solve in this lesson include continuous compounding interest and continuous radioactive decay. y f() = e (, e) 3 E X A M P L E Evaluate f() = e to the nearest thousandth for each value of below. a. = b. = c. = SOLUTION a. f() = e b. f ( ) = e c. f( ) = e TECHNOLOGY GRAPHICS CALCULATOR CHECK Use a table of values for y = e or a graph of y = e to verify your answers. Keystroke Guide, page 0 (, 7.389) (, 0.368) (0.5,.69) TRY THIS Evaluate f() = e to the nearest thousandth for = 6 and = 3. Many banks compound the interest on accounts daily or monthly. However, some banks compound interest continuously, or at every instant, by using the continuous compounding formula,which includes the number e. Continuous Compounding Formula If P dollars are invested at an interest rate, r,that is compounded continuously, then the amount, A,ofthe investment at time t is given by A = Pe rt. LESSON 6.6 THE NATURAL BASE, e 393

38 E X A M P L E APPLICATION INVESTMENTS An investment of $000 earns an annual interest rate of 7.6%. Compare the final amounts after 8 years for interest compounded quarterly and for interest compounded continuously. SOLUTION Substitute 000 for P,0.076 for r, and 8 for t in the appropriate formulas. Compounded quarterly A = P(+ r n ) nt Compounded continuously A = Pe rt A = 000( ) A = 000e A 86.3 A Interest that is compounded continuously results in a final amount that is about $0 more than that for the interest that is compounded quarterly. TRY THIS Find the value of $500 after years invested at an annual interest rate of 9% compounded continuously. The Natural Logarithmic Function CHECKPOINT The natural logarithmic function, y = log e,abbreviated y = ln, is the inverse of the natural eponential function, y = e.the function y = ln is graphed along with y = e at right. State the domain and range of y = e and of y = ln. y y = e y = y = ln E X A M P L E 3 Evaluate f() = ln to the nearest thousandth for each value of below. a. = b. = c. = SOLUTION f() = ln f ( ) = ln a. b. c f( ) = ln( ) is undefined. TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page 0 CHECK Use a table of values for y = ln or a graph of y = ln to verify your answers. Nonpositive numbers are not in the domain of y = ln. (, 0.693) (, 0.693) 39 CHAPTER 6

39 E X A M P L E The natural logarithmic function can be used to solve an equation of the form A = Pe rt for the eponent t in order to find the time it takes for an investment that is compounded continuously to reach a specific amount. This is shown in Eample. How long does it take for an investment to double at an annual interest rate of 8.5% compounded continuously? PROBLEM SOLVING TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page 0 TRY THIS CRITICAL THINKING SOLUTION Use the formula A = Pe rt with r = A = Pe 0.085t P = Pe 0.085t When the investment doubles, A = P. = e 0.085t ln = ln e 0.085t Take the natural logarithm of both sides. ln = 0.085t Use the Eponential-Logarithmic Inverse Property. t = ln t 8.5 CHECK Graph y = e and y =, and find the point of intersection. Notice that the graph of y = e is a horizontal stretch of the function y = e by a factor of 0.0,or almost. 85 Thus, it takes about 8 years and months to double an investment at an annual interest rate of 8.5% compounded continuously. How long does it take for an investment to triple at an annual interest rate of 7.% compounded continuously? Eplain why the time required for the value of an investment to double or triple does not depend on the amount of principal. Radioactive Decay Most of the carbon found in the Earth s atmosphere is the isotope carbon-, but a small amount is the radioactive isotope carbon-. Plants absorb carbon dioide from the atmosphere, and animals obtain carbon from the plants they consume. When a plant or animal dies, the amount of carbon- it contains decays in such a way that eactly half of its initial amount is present after 5730 years. The function below models the decay of carbon-, where N 0 is the initial amount of carbon- and N(t) is the amount present t years after the plant or animal dies. N(t) = N 0 e 0.000t LESSON 6.6 THE NATURAL BASE, e 395

40 E X A M P L E 5 APPLICATION ARCHAEOLOGY Eample 5 shows how radiocarbon dating is used to estimate the age of an archaeological artifact. Suppose that archaeologists find scrolls and claim that they are 000 years old. Tests indicate that the scrolls contain 78% of their original carbon-. Could the scrolls be 000 years old? SOLUTION Since the scrolls contain 78% of their original carbon-, substitute 0.78N 0 for N(t). N(t) = N 0 e 0.000t 0.78N 0 = N 0 e 0.000t Substitute 0.78N 0 for N(t) = e 0.000t ln 0.78 = 0.000t Take the natural logarithm of each side t = ln 0.78 t = ln t Thus, it appears that the scrolls are about 000 years old. Eercises Communicate. Compare the natural and eponential logarithmic functions with the base-0 eponential and logarithmic functions.. Give a real-world eample of an eponential growth function and of an eponential decay function that each have the base e. 3. State the continuous compounding formula, and describe what each variable represents.. Describe how the continuous compounding formula can represent continuous growth as well as continuous decay. Guided Skills Practice A P P L I C A T I O N Evaluate f() = e to the nearest thousandth for each value of. (EXAMPLE ) 5. = 3 6. = INVESTMENTS An investment of $500 earns an annual interest rate of 8.%. Compare the final amounts after 5 years for interest compounded quarterly and for interest compounded continuously. (EXAMPLE ) 396 CHAPTER 6

41 Evaluate f() = ln to the nearest thousandth for each value of. (EXAMPLE 3) 8. = 5 9. =.5 APPLICATIONS 0. INVESTMENTS How long does it take an investment to double at an annual interest rate of 7.5% compounded continuously? (EXAMPLE ). ARCHAEOLOGY A piece of charcoal from an ancient campsite is found in an archaeological dig. It contains 9% of its original amount of carbon-. Estimate the age of the charcoal. (EXAMPLE 5) Homework Help Online Go To: go.hrw.com Keyword: MB Homework Help for Eercises 3 Practice and Apply Evaluate each epression to the nearest thousandth. If the epression is undefined, write undefined.. e 6 3. e 9. e. 5. e e e e e e. e. ln 3 3. ln 7. ln 0,00 5. ln 99, ln ln ln 5 9. ln ln( ) 3. ln( 3) For Eercises 3 35, write the epressions in ascending order. 3. e, e 5,ln, ln e, e 0,ln, ln 3. e.5,ln.5, 0.5,log e.3,ln.3, 0.3,log.3 State whether each equation is always true, sometimes true, or never true. 36. e 5 e 3 = e (e ) 3 = e 38. e 6 = e 6 e 39. e = e e Simplify each epression. 0. e ln. e ln 5. e 3ln 3. e ln 5. ln e 3 5. ln e 6. 3 ln e 7. ln e Write an equivalent eponential or logarithmic equation. 8. e = e = 50. ln ln e e C H A L L E N G E Solve each equation for by using the natural logarithm function. Round your answers to the nearest hundredth = = = = = = Sketch f() = e for. A line that intersects a curve at only one point is called a tangent line of the curve. a. Sketch lines that are tangent to the graph of f() = e at = 0.5, = 0, =, and =. b. Find the approimate slope of each tangent line. Compare the slope of each tangent line with the corresponding y-coordinate of the point where the tangent line intersects the graph. c. Make a conjecture about the slope of f() = e as increases. LESSON 6.6 THE NATURAL BASE, e 397

42 CONNECTIONS TRANSFORMATIONS Let f() = e. For each function, describe the transformations from f to g. 6. g() = 6e + 6. g() = 0.75e 63. g() = 0.5e ( + ) ( ) 6. g() = 3e f() = e g() = e i() = e h() = e TRANSFORMATIONS Let f() = ln. For each function, describe the transformations from f to g. 65. g() = 3 ln( + ) 66. g() = ln( ) 67. g() = 0.5 ln(5) 68. g() = 5 ln(0.5) TRANSFORMATIONS The graphs of f() = e, g() = e, h() = e, and i() = e are shown on the same coordinate plane at left. What transformations relate each function, f, g, and i,to h? 70. TRANSFORMATIONS For f() = e,describe how each transformation affects the domain, range, asymptotes, and y-intercept. a. a vertical stretch b. a horizontal stretch c. a vertical translation d. a horizontal translation 7. TRANSFORMATIONS For f() = ln,describe how each transformation affects the domain, range, asymptotes, and -intercept. a. a vertical stretch b. a horizontal stretch c. a vertical translation d. a horizontal translation APPLICATIONS 7. PHYSICS The amount of radioactive strontium-90 remaining after t years decreases according to the function N(t) = N 0 e 0.038t.How much ofa 0-gram sample will remain after 5 years? 73. ECONOMICS The factory sales of pagers from 990 through 995 can be modeled by the function S = 6e 0.8t,where t = 0 in 990 and S represents the sales in millions of dollars. [Source: Electronic Market Data Book] a. According to this function, find the factory sales of pagers in 995 to the nearest million. b. If the sales of pagers continued to increase at the same rate, when would the sales be double the 995 amount? 7. INVESTMENTS Compare the growth of an investment of $000 in two different accounts. One account earns 3% annual interest, the other earns 5% annual interest, and both are compounded continuously over 0 years. 75. ARCHAEOLOGY A wooden chest is found and is said to be from the second century B.C.E. Tests on a sample of wood from the chest reveal that it contains 9% of its original carbon-. Could the chest be from the second century B.C.E.? 398 CHAPTER 6

43 Sales (in millions of $) Basketball Backboard Sales BUSINESS Sales of home basketball backboards from 986 to 996 can be modeled by S = 0.65e 0.57t,where S is the sales in millions of dollars, t is time in years, and t = 0 in 986. [Source: Huffy Sports] a. Use this model to estimate the sales of backboards in 997 to the nearest thousand. b. If the sales of basketball backboards continued to increase at the same rate,when would the sales of basketball backboards be double the amount of 996? INVESTMENTS For Eercises 77 79, assume that all interest rates are compounded continuously. 77. How long will it take an investment of $5000 to double if the annual interest rate is 6%? 78. How long will it take an investment to double at 0% annual interest? 79. If it takes a certain amount of money 3.7 years to double, at what annual interest rate was the money invested? 80. AGRICULTURE The percentage of farmers in the United States workforce has declined since the turn of the century. The percent of farmers in the workforce, f, can be modeled by the function f(t) = 9e 0.036t,where t is time in years and t = 0 in 90. Find the percent of farmers in the workforce in 995. [Source: Bureau of Labor Statistics] Look Back Solve each inequality, and graph the solution on a number line. (LESSON.8) Graph each function. (LESSON.7) 8. f() = 85. g() = 86. h() = [ 3] Graph each system. (LESSON 3.) 87. y > > y y + 3y 8 y + 0 > y 3 Factor each epression. (LESSON 5.3) Look Beyond 93 Solve ln + ln( + ) = 5 by graphing y = ln + ln( + ) and y = 5 and finding the -coordinate of the point of intersection. LESSON 6.6 THE NATURAL BASE, e 399

44 Solving Equations and Modeling Objectives Solve logarithmic and eponential equations by using algebra and graphs. Model and solve realworld problems involving eponential and logarithmic relationships. Why Physicists, chemists, and geologists use eponential and logarithmic equations to model various phenomena, such as the magnitude of earthquakes. Magnitude RICHTER SCALE RATINGS Result near the epicenter Approimate number of occurrences per year 8 9 near total damage serious damage to buildings moderate damage to buildings slight damage to buildings felt by most people felt by some people 6, not felt but recorded 800,000 APPLICATION GEOLOGY On the Richter scale, the magnitude, M, ofan earthquake depends on the amount of energy, E,released by the earthquake as follows: M = 3 log E.8 0 The amount of energy, measured in ergs, is based on the amount of ground motion recorded by a seismograph at a known distance from the epicenter of the quake. The logarithmic function for the Richter scale assigns very large numbers for the amount of energy, E, to numbers that range from to 9. A rating of on the Richter scale indicates the smallest tremor that can be detected. Destructive earthquakes are those rated greater than 6 on the Richter scale. 0 CHAPTER 6

45 E X A M P L E APPLICATION GEOLOGY PROBLEM SOLVING One of the strongest earthquakes in recent history occurred in Meico City in 985 and measured 8. on the Richter scale. Find the amount of energy, E, released by this earthquake. SOLUTION Use a formula. M = 3 log E.8 0 A seismogram produced by a seismograph 8. = 3 log E.8 Substitute 8. for the magnitude, M. 0.5 = log E = E 0.8 Use the definition of logarithm = E E Write the answer in scientific notation. The amount of energy, E,released by this earthquake was approimately ergs. In physics, an erg is a unit of work or energy. To solve the logarithmic equation in Eample, you must use the definition of a logarithm. However, solving eponential and logarithmic equations often requires a variety of the definitions and properties from this chapter. A summary of the definitions and properties that you have learned is given below. Copy these properties and definitions into your notebook for reference. SUMMARY Eponential and Logarithmic Definitions and Properties Definition of logarithm y = log b if and only if b y = Product Property log b mn = log b m + log b n Quotient Property log b ( m n ) = log b m log b n Power Property log b m p = p log b m Eponential-Logarithmic b log b = for > 0 Inverse Properties log b b = for all One-to-One Property of Eponents One-to-One Property of Logarithms Change-of-base formula If b = b y, then = y. If log b = log b y, then = y. logb a log c a = log c b CHECKPOINT Show how to solve M = 3 log E.8 for E. 0 CRITICAL THINKING Use the properties of eponents and logarithms to show that log a( ) = log a. LESSON 6.7 SOLVING EQUATIONS AND MODELING 03

46 E X A M P L E Solve log + log( 3) = for. SOLUTION Method Use algebra. log + log( 3) = log[( 3)] = Apply the Product Property of Logarithms. ( 3) = 0 Write the equivalent eponential equation. 3 0 = 0 ( 5)( + ) = 0 = 5 or = TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page CHECK Let = 5. Let =. log + log( 3) = log + log( 3) = log 5 + log =? log( ) + log( 5) = Undefined = True Since the domain of a logarithmic function ecludes negative numbers, the only solution is 5. Method Use a graph. Graph y = log + log( 3) and y =, and find the point of intersection. The coordinates of the point of intersection are (5, ), so the solution is 5. TRY THIS Solve log( + 8) + log = by using algebra and a graph. E X A M P L E 3 Solve e 3 5 = 7 for. SOLUTION Method Use algebra. e 3 5 = 7 e 3 5 = 8 ln e 3 5 = ln = ln 8 = ln Take the natural logarithm of each side. Use Eponential-Logarithmic Inverse Properties. Eact solution Approimate solution TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page Method Use a graph. Graph y = e 3 5 and y = 7, and find the point of intersection. The coordinates of the point of intersection are approimately (.63, 7), so the solution is approimately CHAPTER 6

47 TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page CHECKPOINT CHECKPOINT Solving Eponential Inequalities You will need: a graphics calculator. Graph y = log + log( + ) and y = on the same screen.. For what value(s) of is y = y? y < y? y > y? 3. Eplain how you can use a graph to solve log + log( + ) >.. Graph y = e and y = 38 on the same screen. 5. For what approimate value(s) of is y = y? y < y? y > y? 6. Eplain how you can use a graph to solve e < 38. Newton s Law of Cooling An object that is hotter than its surroundings will cool off, and an object that is cooler than its surroundings will warm up. Newton s law of cooling states that the temperature difference between an object and its surroundings decreases eponentially as a function of time according to the following: T(t) = T s + (T 0 T s )e kt T 0 is the initial temperature of the object, T s is the temperature of the object s surroundings (assumed to be constant), t is the time, and k represents the constant rate of decrease in the temperature difference (T 0 T s ). E X A M P L E APPLICATION PHYSICS When a container of milk is taken out of the refrigerator, its temperature is 0 F. An hour later, its temperature is 50 F. Assume that the temperature of the air is a constant 70 F. a. Write the function for the temperature of this container of milk as a function of time, t. b. What is the temperature of the milk after hours? c. After how many hours is the temperature of the milk 65 F? SOLUTION a. First substitute 0 for T 0 and 70 for T s, and simplify. T(t) = T s + (T 0 T s )e kt T(t) = 70 + (0 70)e kt T(t) = 70 + ( 30)e kt Since T() = 50, substitute for t and 50 for T(t), and solve for k. 50 = 70 30e k 30e k = 0 e k = 3 ln e k = ln 3 k = ln 3 LESSON 6.7 SOLVING EQUATIONS AND MODELING 05

48 TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page Substitute ln 3 for k and simplify to get the function for the temperature of this container of milk. T(t) = 70 30e kt T(t) = 70 30e (ln 3 )t T(t) = ( 3 ) t Apply the Eponential-Logarithmic Inverse Property. The function for the temperature of this container of milk is T(t) = ( 3 ) t. b. Find T(). T(t) = ( 3 ) t T() = ( 3 ) 56.7 The temperature of the milk after hours is approimately 56.7 F. c. Substitute 65 for T(t), and solve for t. T(t) = ( 3 ) t 65 = ( 3 ) t 30 ( 3) t = 5 ( 3) t = 6 t ln 3 = ln 6 ln 6 t =. ln 3 It will take approimately. hours, or about hours and 5 minutes, for the milk to warm up to 65 F. Eercises Activities Online Go To: go.hrw.com Keyword: MB Spacecraft Communicate. Eplain how to solve the eponential equation e +7 = 98 by algebraic methods.. How can you solve the logarithmic equation log + log ( + 3) = by algebraic methods? 3. Eplain how to solve eponential and logarithmic equations by graphing. 06 CHAPTER 6

49 APPLICATIONS Guided Skills Practice. GEOLOGY In 989, an earthquake that measured 7. on the Richter scale occurred in San Francisco, California. Find the amount of energy, E, released by this earthquake. (EXAMPLE ) 5. Solve log( 90) + log = 3 for. (EXAMPLE ) 6. Solve 0.5e 0.08t = 0 for. (EXAMPLE 3) 7. PHYSICS When the air temperature is a constant 70 F, an object cools from 70 F to 0 F in one-half hour. (EXAMPLE ) a. Write the function for the temperature of this object, T,as a function of time, t. b. What is the temperature of this object after hour? c. After how many hours is the temperature of this object 90 F? Homework Help Online Go To: go.hrw.com Keyword: MB Homework Help for Eercises 8 5 C H A L L E N G E APPLICATIONS Practice and Apply Solve each equation for. Write the eact solution and the approimate solution to the nearest hundredth, when appropriate = = = 5. = log 3 7. = log 6 3. log = 6. = log 6 5. e = 0 6. e (+) = 7. ln( 3) = ln 8. ln( + 3) = ln = e = ln = ln + ln. ln + ln( + ) = ln 3. ln + =. 3 ln + 3 = 5. 3 log + 7 = 5 Solve each equation for. Write the eact solution and the approimate solution to the nearest hundredth, when appropriate. 6. ln(3 ) = ln 7. log 3 = (log ) 3 8. GEOLOGY On May 0, 997, a light earthquake with a magnitude of.7 struck the Calaveras Fault 0 miles east of San Jose, California. Find the amount of energy, E,released by this earthquake. 9. GEOLOGY In 976, an earthquake that released about ergs of energy occurred in San Salvador, El Salvador. Find the magnitude, M, of this earthquake to the nearest tenth. 30. PHYSICS A hot coal (at a temperature of 60 C) is immersed in ice water (at a temperature of 0 C). After 30 seconds, the temperature of the coal is 60 C. Assume that the ice water is kept at a constant temperature of 0 C. a. Write the function for the temperature, T,ofthis object as a function of time, t, in seconds. b. What will be the temperature of this coal after minutes (0 seconds)? c. After how many minutes will the temperature of the coal be C? Temperature ( C) Cooling Temperature of a Hot Coal Time (s) LESSON 6.7 SOLVING EQUATIONS AND MODELING 07

50 APPLICATIONS 6.8 magnitude earthquake off the coast of northern California on February 9, GEOLOGY Compare the amounts of energy released by earthquakes that differ by in magnitude. In other words, how much more energy is released by an earthquake of magnitude 6.8 than an earthquake of magnitude 5.8? The map shows the locations of the three strongest earthquakes in the United States in magnitude earthquake in Ridgecrest, California, northeast of Los Angeles on September 0, magnitude earthquake in Brewster County, Teas, on April, PSYCHOLOGY Educational psychologists sometimes use mathematical models of memory. Suppose that some students take a chemistry test. After a time, t (in months), without a review of the material, they take an equivalent form of the same test. The mathematical model a(t) = 8 log(t + ), where a is the average score at time t, is a function that describes the students retention of the material. a. What is the average score when the students first took the test (t = 0)? b. What is the average score after 6 months? c. After how many months is the average score 60? 33. BIOLOGY A population of bacteria grows eponentially. A population that initially consists of 0,000 bacteria grows to 5,000 bacteria after hours. a. Use the eponential growth function, P(t) = P 0 e kt,to find the value of k.then write a function for this population of bacteria in terms of time, t.round the value of k to the nearest hundredth. b. How many bacteria will the population consist of after hours, rounded to the nearest hundred thousand? c. How many bacteria will the population consist of after hours? 3. DEMOGRAPHICS The population of India was estimated to be 57,0,000 in 97 and 76,388,000 in 98. Assume that this population growth is eponential. Let t = 0 represent 97 and t = 0 represent 98. a. Use the eponential growth function, P(t) = P 0 e kt,to find the value of k. Then write the function for this population as a function of time, t. b. Estimate the population in 00, rounded to the nearest hundred thousand. c. Use the function you wrote in part a to estimate the year in which the population will reach.5 billion. 35. ARCHEOLOGY Refer to the discussion of radioactive decay on page 395. Suppose that an animal bone is unearthed and it is determined that the amount of carbon- it contains is 0% of the original amount. a. Use the decay function for carbon-, N(t) = N 0 e 0.000t,to write an equation using the percentage of carbon- given above. b. Use the equation you wrote in part a to find the approimate age, t,of the bone. 08 CHAPTER 6

51 Look Back Graph each system of linear inequalities. (LESSON.8) 5y < 3 y y 5 > 3 y + 3 < y + 8 Solve each equation for. (LESSON 5.) = = Solve for, and check your answers. (LESSON 6.). log b ( ) = log b ( + ). log 0 (8 + ) = log 0 ( 8) 3. log a ( + ) + log a = log a 0. log b log b 3 = log b ( 3) A P P L I C A T I O N INVESTMENTS Assume that all interest rates are compounded continuously in Eercises 5 7. (LESSON 6.6) 5. How long will it take an investment of $5000 to double if the annual interest rate is 5%? 6. How long will it take an investment to double at 8% annual interest? 7. If it takes a certain amount of money 3. years to double, at what annual interest rate was the money invested? Portfolio Etension Go To: go.hrw.com Keyword: MB Newton Look Beyond 8. Graph each function and compare the shapes of the graphs. a. y = b. y = 3 c. y = PORTFOLIO A C T I T Y I V This activity requires the data collected for the Portfolio Activities on pages 36, 369, and 38.. Refer to your data from the Portfolio Activity on page 36. a.use Newton s law of cooling, found in Eample on page 05, to write a function that models the temperature of the probe as it cooled to the temperature of ice water. b. Compare the graph of this function with the graph of the eponential function that you generated for the same data in the Portfolio Activity on page 369. (Hint: You can use the table function on your graphics calculator to compare the y-values of these functions with the original values.). a. Repeat part a of Step, using the data collected in the Portfolio Activity on page 38. b. Repeat part b from Step, comparing your new graph with the graphs of both the eponential function and the approimating function from the Portfolio Activity on page 38. WORKING ON THE PROJECT You should now be able to complete the Chapter Project. LESSON 6.7 SOLVING EQUATIONS AND MODELING 09

52 Chapter Review and Assessment VOCABULARY asymptote base change-of-base formula common logarithm compound interest formula continuous compounding formula effective yield eponential decay eponential epression eponential function eponential growth Eponential-Logarithmic Inverse Properties logarithmic function multiplier natural base natural eponential function natural logarithmic function Newton s law of cooling One-to-One Property of Eponents One-to-One Property of Logarithms Power Property of Logarithms Product Property of Logarithms Quotient Property of Logarithms Key Skills & Eercises LESSON 6. Key Skills Write and evaluate eponential epressions. The world population rose to about 5,73,000,000 in 995. The world population was increasing at an annual rate of.6%. Write and evaluate an epression to predict the world population in 00. [Source: Worldbook Encyclopedia] 5,73,000,000(.06) 5,73,000,000(.06) 5 8,57,000,000 The projected world population for 00 is about 8.5 billion people. Eercises. INVESTMENTS The value of a painting is $,000 in 990 and increases by 8% of its value each year. Write and evaluate an epression to estimate the painting s value in DEPRECIATION The value of a new car is $3,000 in 998; it loses 5% of its value each year. Write and evaluate an epression to estimate the car s value in 005. LESSON 6. Key Skills Classify an eponential function as eponential growth or eponential decay. When b >, the function f() = b represents eponential growth. When 0 < b <, the function f() = b represents eponential decay. Calculate the growth of investments. The total amount of an investment, A, earning compound interest is A(t) = P ( + n r ) nt,where P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years. Eercises Identify each function as representing eponential growth or decay. 3. f() = (0.89). g() = 3 (.06) 5. h() = 5(.06) 6. j() = 5 ( 5 ) INVESTMENTS For each compounding period below, find the final amount of a $00 investment after years at an annual interest rate of.5%. 7. annually 8. quarterly 9. daily CHAPTER 6

53 LESSON 6.3 Key Skills Write equivalent forms of eponential and logarithmic equations. 3 = 8 is log 3 8 = in logarithmic form. log 6 = 6 is 6 = 6 in eponential form. Use the definitions of eponential and logarithmic functions to solve equations. v = log = log v = log v 8 6 v = 36 3 = v v = 8 6 v = 6 6 = v v = 3 v = v = 3 LESSON 6. Key Skills Use the Product, Quotient, and Power Properties of Logarithms to simplify and evaluate epressions involving logarithms. Given log , log 3 63 can be approimated as shown below. log 3 63 = log log 3 7 = log = 7 log 5 5 = 7 = Eercises 0. Write 5 = 5 in logarithmic form.. Write log 3 7 = 3 in eponential form.. Write log 3 = in eponential form. 9 Find the value of v in each equation. 3. v = log 8 6. log v = 5. = log v 6. 3 = log v log v = 3 8. log 7 3 = v 9. log v 9 = 0. log 6 = v Eercises Given log and log 7 9.9, approimate the value of each logarithm.. log 7 5. log log 7 35 Write each epression as a single logarithm. Then simplify, if possible.. log log log log 6 log 3 + log 7 Evaluate each epression. 6. log 7. log log LESSON 6.5 Key Skills Use the common logarithmic function to solve eponential and logarithmic equations. = log = 3 0 = log = log 3 0,000 = log = log 3 = l og 3 log 5.09 Apply the change-of-base formula to evaluate logarithmic epressions. logb The change-of-base formula is log a = l, ogb a where a, b, and > 0. log 5 = l og 5.3 log Eercises Solve each equation. Give your answers to the nearest hundredth. 9. log = log 0.0 = = = = = 36 Evaluate each logarithmic epression to the nearest hundredth. 35. log log log log CHEMISTRY What is [H + ] of a carbonated soda if its ph is.5? CHAPTER 6 REVIEW 3

54 LESSON 6.6 Key Skills Evaluate eponential functions of base e and natural logarithms. Using a calculator and rounding to the nearest thousandth, e.5.8 and ln Model eponential growth and decay processes by using base e. The continuous compounding formula is A = Pe rt, where A is the final amount when the principal P is invested at an annual interest rate of r for t years. Eercises Evaluate each epression to the nearest thousandth. 0. e 0.5. e 5. ln 5 3. ln INVESTMENTS Sharon invests $500 at an annual interest rate of 9%. How much is the investment worth after 0 years if the interest is compounded continuously? LESSON 6.7 Key Skills Solve logarithmic and eponential equations. log 3 = 5 ln = 5 = 3 3 ln = 5 = 5 ln = 3 = ln( + 6) = ln 3 ln( + 6) = ln = 9 = 3 = e Eercises Solve each equation for. Write the eact solution and the approimate solution to the nearest hundredth, when appropriate. 5. log = ln() = ln 7. log = log ln 3 = 9. HEALTH The normal healing of a wound can be modeled by A = A 0 e 0.35n,where A is the area of the wound in square centimeters after n days. After how many days is the area of the wound half of its original size, A 0? Applications 50. BIOLOGY Given favorable living conditions, fruit fly populations can grow at the astounding rate of 8% per day. If a laboratory selects a population of 5 fruit flies to reproduce, about how big will the population be after 3 days? after 5 days? after week? 5. PHYSICS Suppose that the sound of busy traffic on a four-lane street is about times the intensity of the threshold of hearing, I 0.Find the relative intensity, R,in decibels of the traffic on this street. 5. PHYSICS Radon is a radioactive gas that has a half-life of about 3.8 days. This means that only half of the original amount of radon gas will be present after about 3.8 days. Using the eponential decay function A = Pe kt, find the value of k to the nearest hundredth, and write the function for the amount of radon remaining after t days. Fruit fly CHAPTER 6

55 Chapter Test. DEMOGRAPHICS The population of Petoskey, Michigan, was 6076 in 990 and was growing at the rate of 3.7% per year. The city planners want to know what the population will be in the year 05. Write and evaluate an epression to estimate this population.. INCOME TAX The government allows for linear depreciation of capital ependitures for income ta purposes at the rate of 0% per year. What will be the value of a $50,000 tool and die machine after 7 years of use? Tell whether each function represents eponential growth or decay. 3. f() 3.6(.0). g(t) 0.05(.3) t 5. h(t) t 6. j() 500(0.5) INVESTMENTS For each compounding period below, find the final amount of a $5000 investment after 0 years at a 5.6% annual interest rate. 7. daily 8. monthly 9. quarterly 0. annually Write each logarithmic equation in eponential form and each eponential equation in logarithmic form.. log log 5 Find the value of v in each equation log v log 9 79 v 7. log 6 v 5 65 Write each epression as a single logarithm. Then simplify, if possible. 8. log 5 3log 3 + log 6 9. log 7 + log 7 log 76 Evaluate each epression log 5 3. log log log b b ( ) Solve each equation. Give your answers to the nearest hundredth.. log log SEISMOLOGY The amount of energy E,in ergs, released by an earthquake of magnitude M is given by the formula E 0 (.5M.8).What is the difference in the amount of energy released by an earthquake of magnitude 6.5 and one of magnitude 8.7? Evaluate each epression to the nearest thousandth. 9. e lnπ 3. e ln(e.68 ) 33. ARCHAEOLOGY The age of an artifact can be determined using carbon- dating with the equation N(t) N 0 e t.What is the approimate age of an artifact if a sample reveals that it contains 3% of its original carbon-? Solve each equation for. Write the eact solution and the approimate solution to the nearest hundredth, when appropriate ln( + ) ln 36. log + log( + 3) CHAPTER 6 TEST 5