Chapter 3 A1 Glencoe Precalculus DATE Before you begin Chapter 3 Exponential and Logarithmic Functions Anticipation Guide PERIOD A D D 2. Logarithmic functions and exponential functions are inverses of each other. 3. A logarithm with base 10 or log 10 is called a natural logarithm. 4. A logarithm with base e or log e is called a common logarithm and is denoted by ln. D 8. Exponential models apply to any situation where the change is proportional to the initial size of the quantity being considered. After you complete Chapter 3 A 7. A logistic function is shown on a graph that changes rapidly and then has a horizontal asymptote. 6. Logarithmic functions do not have the one-to-one property. A D A 1. For most real-world applications involving exponential functions, the most commonly used base is not 2 or 10 but an irrational number e. 5. Exponential functions have the one-to-one property. STEP 2 A or D Statement Chapter 3 3 Chapter Resources 3/18/09 5:18:44 PM Answers Glencoe Precalculus • For those statements that you mark with a D, use a piece of paper to write an example of why you disagree. • Did any of your opinions about the statements change from the first column? • Reread each statement and complete the last column by entering an A or a D. Step 2 STEP 1 A, D, or NS • Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure). • Decide whether you Agree (A) or Disagree (D) with the statement. • Read each statement. Step 1 3 NAME 0ii_004_PCCRMC03_893804.indd Sec1:3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. DATE PERIOD 27 -3 9 -2 3 -1 1 0 1 − 9 1 − 3 27 1 − 3 Asymptote: x-axis Range: (0, ∞) 2 1 x→∞ x 0 y ( x 2 ) 5 decreasing: (-∞, ∞) increasing: (-∞, ∞) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 5 Chapter 3 x → -∞ lim f(x) = 0; x→∞ x → -∞ lim f(x) = ∞; x→∞ x asymptotes: y = 1; end behavior: f(x) = 1 and lim 0 y 2. k(x) = e-2x D = (-∞, ∞); R = (0, ∞); intercept: (0, 1); asymptote: x-axis; end behavior: lim f(x) = ∞ and D = (-∞, ∞); R = (1, ∞); 1 intercept: 0, 1 − ; 0 y 1. h(x) = 2x - 1 + 1 x Lesson 3-1 10/1/09 1:42:35 PM Glencoe Precalculus Sketch and analyze the graph of each function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. Exercises Decreasing: (-∞, ∞) x → -∞ End behavior: lim f(x) = ∞ and lim f(x) = 0 Intercept: (0, 1) Domain: (-∞, ∞) f(x) x () 1 . Describe Sketch and analyze the graph of f(x) = − 3 its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. Example An exponential function with base b has the form f(x) = abx, where x is any real number and a and b are real number constants such that a ≠ 0, b is positive, and b ≠ 1. If b > 1, then the function is exponential growth. If 0 < b < 1, then the function is exponential decay. Exponential Functions Study Guide and Intervention Exponential Functions 3-1 NAME Answers (Anticipation Guide and Lesson 3-1) Exponential Functions Study Guide and Intervention DATE (continued) PERIOD N = N0ekt N is the final amount, N0 is the initial amount, r is the rate of growth or decay, r is time, and e is a constant. N = N0(1 + r)t N is the final amount, N0 is the initial amount, r is the rate of growth or decay, and t is time. nt P is the principal or initial investment, A is the final amount of the investment, r is the annual interest rate, n is the number of times interest is compounded each year, and t is the number of years. r A = P1 + − n Compound Interest Use a calculator. N0 = 100, r = 0.25, t = 6 Exponential Growth Formula A2 Glencoe Precalculus 005_032_PCCRMC03_893804.indd 6 Chapter 3 6 Glencoe Precalculus 3. BIOLOGY The number of rabbits in a field showed an increase of 10% each month over the last year. If there were 10 rabbits at this time last year, how many rabbits are in the field now? 33 rabbits quadrillion thermal units 2. ENERGY In 2007, it is estimated that the United States used about 101,000 quadrillion thermal units. If U.S. energy consumption increases at a rate of about 0.5% annually, what amount of energy will the United States use in 2020? about 107,766 with continuous 1. FINANCIAL LITERACY Compare the balance after 10 years of a $5000 investment earning 8.5% interest compounded continuously to the same investment compounded quarterly. continuously: $11,698.23, quarterly $11,594.52, $103.71 more Exercises Example 2 FINANCIAL LITERACY Lance has a bank account that will allow him to invest $1000 at a 5% interest rate compounded continuously. If there are no other deposits or withdrawals, what will Lance’s account balance be after 10 years? A = Pert Continuous Compound Interest Formula = 1000e(0.05)(10) P = 1000, r = 0.05, and t = 10 ≈ 1648.72 Simplify. With continuous compounding, Lance’s account balance after 10 years will be $1648.72. There will be about 381 cells in the colony in 6 weeks. N = N0(1 + r)t = 100(1 + 0.25)6 ≈ 381.4697266 Example 1 BIOLOGY A researcher estimates that the initial population of a colony of cells is 100. If the cells reproduce at a rate of 25% per week, what is the expected population of the colony in six weeks? Continuous Exponential Growth or Decay Exponential Growth or Decay 10/1/09 1:43:46 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 3 Exponential Growth and Decay Many real-world situations can be modeled by exponential functions. One of the equations below may apply. 3-1 NAME Exponential Functions Practice DATE PERIOD 4 −4 0 4 8x decreasing: (-∞, ∞) and Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 7 Chapter 3 7 4. FINANCE Determine the amount of money in a savings account that provides an annual rate of 4% compounded monthly if the initial deposit is $1000 and the money is left in the account for 5 years. $1221.00 b. Predict the population in 2020 and 2030. Assume a steady rate of increase. 335,319,934 in 2020 and 366,024,459 in 2030 Lesson 3-1 3/18/09 6:00:31 PM Glencoe Precalculus a. Let t be the number of years since 2000. Write a function that models the annual growth in population in the U.S. N = 281421906(1 + 0.0088)t 3. DEMOGRAPHICS In 2000, the number of people in the United States was 281,421,906. The U.S. population is estimated to be growing at 0.88% annually. increasing: (-∞, ∞) and f(x) = -2 lim x → -∞ lim f(x) = -∞; x→∞ asymptote: y = -2; asymptote: x-axis; end behavior: x-intercept: none; x-intercept: none; f(x) = 0 lim x → -∞ lim f(x) = ∞; x→∞ y-intercept: (0, -2.2); end behavior: R = (-∞, -2); R = (0, ∞); 1 ; y-intercept: 0, − ( 2) y 5 D = (-∞, ∞); −8 −8 −8 8x −4 4 4 8 1 x 2. h(x) = - − e -2 −4 −4 0 y D = (-∞, ∞); −8 8 1. f(x) = 2 x - 1 Sketch and analyze the graph of each function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. 3-1 NAME Answers (Lesson 3-1) Chapter 3 Exponential Functions A3 8 PERIOD 215 210 220 2011 225 2021 230 2031 Glencoe Precalculus 3/18/09 6:00:37 PM Answers Glencoe Precalculus No; week 21 has 220 seconds of homework or about 291 hours of homework. 5. If your precalculus teacher offers to give you 1 second of homework for the first week of school and double the amount of homework each week until the end of the school year (i.e. 2 seconds the second week), should you say yes? Explain. 2001 1991 −t P = 1 22 c. A 1971 computer could manage one process per second. Every two years, the number of processes also doubles on a computer. Write an equation to calculate the number of processes a computer can manage every second in each year after 1971. Then complete the table below. 1.1 × 109 b. Approximately how many transistors were on one circuit in 2009? N = 2100 2 2 −t a. If there were about 2100 transistors on every circuit in 1971, write an exponential equation to model the number of transistors in a given year t after 1971. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 8 Chapter 3 Sample answer: I would choose the higher monthly rate. To check, the balance is $1120.45 versus $1118.51; after 5 years, the balance is $1328.87 versus $1323.13. 3. FINANCAL LITERACY You have $1000 to put into the bank. One bank offers a 5.7% interest rate compounded monthly. Another bank offers 5.6% compounded continuously. Which would you choose to make the most money after 2 years? after 5 years? Explain. P(5) = 388; P(10) = 1503; P(24) = 66,781; P(72) = 2.98 × 1010 c. Find the number of bacteria in 5, 10, 24, and 72 hours. Round to the nearest whole number. 100 b. What was the initial number of bacteria? 27.1% per hour a. Determine the growth rate. 2. BIOLOGY Suppose a certain type of bacteria reproduces according to the model P(t) = 100e0.271t, where t is time in hours and P(t) is the number of bacteria. $1578.18 DATE 4. TECHNOLOGY In 1965, Gordon Moore stated that since the invention of the integrated circuit in 1958, the number of transistors that can be placed on that circuit has doubled every two years. This statement has been true to the present day. Almost all measures of computing power come from this statement so that we can say that the computing power doubles nearly every two years. Word Problem Practice 1. FINANCIAL LITERACY Suppose Jamal has a savings account with a balance of $1400 at a 4% interest rate compounded monthly. If there are no other deposits or withdrawals, what will be Jamal’s account balance in three years? 3-1 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Enrichment DATE PERIOD 2 3 3! 3 3! 2 2 2! 4 4 5 5! 3 3! 2 4 5 5! n n! 2 3 3! 3 3! 2 2! 4 4 4! 5 4! 5 5! 6 6! Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 9 Chapter 3 1. e1.5 4.48 2. e2 7.38 9 3. e2.1 8.16 Lesson 3-1 10/1/09 10:40:06 AM Glencoe Precalculus 4. e2.2 9.02 Use the Maclaurin polynomials to approximate each of the following powers of e to the hundredths place. Exercises 3 3! 2 2! 4 5! 1 1 1 1 1 +− +− +− +− ≈ 2.7181 e1 ≈ 1 + 1 + − 4! 1 1 1 1 +− +− +− ≈ 2.7166 e1 ≈ 1 + 1 + − 3 3! 2 2! 1 1 1 +− +− ≈ 2.7083 e1 ≈ 1 + 1 + − 2! 1 1 +− ≈ 2.6667 e1 ≈ 1 + 1 + − 2 2! 1 = 2.5 e1 ≈ 1 + 1 + − e1 ≈ 1 + 1 = 2 When x = 1, these polynomials can be used to approximate e1 or e to the thousandths place. Remember e ≈ 2.718. 4! x x x x x +− +− +− +…+− . ex ≈ 1 + x + − 2! By adding more and more terms, you get the general Maclaurin polynomial 4! x x x x +− +− +− ex ≈ 1 + x + − 4! x x x +− +− ex ≈ 1 + x + − 2! 3 3! 2 2! x x +− ex ≈ 1 + x + − 2! x ex ≈ 1 + x + − ex ≈ 1 + x It can be shown using calculus that the function e x can be approximated by what are known as Maclaurin polynomials as follows. Maclaurin Polynomials 3-1 NAME Answers (Lesson 3-1) Logarithmic Functions Study Guide and Intervention DATE exponent -2 A4 25 Equality Prop. of Exponents 1 − = 5-2 25 Write in exponential form. 1 Let log5 − = y. Evaluate each logarithm. ln ex = x -4 eln x = x 1 2 2 x 2 6. e ln x 2 3. 3 2 log3 2 10log 13 = 13 c. 10log 13 . because 3 = √3 − 10log x = x Equality Prop. of Exponents 1 − 3 2 = √ 3 Write in exponential form. Let log3 √ 3 = y. 1 =− Therefore, log3 √3 2 exponent Glencoe Precalculus 005_032_PCCRMC03_893804.indd 10 Chapter 3 10 Glencoe Precalculus 7. FINANCIAL LITERACY Ms. Dasilva has $3000 to invest. She would like to invest in an account that compounds continuously at 6%. Use the formula ln A - ln P = rt, where A is the current balance, P is the original principal, r is the rate as a decimal, and t is the time in years. How long will it take for her balance to be $6000? 11.55 years 2 1 5. log3 − 81 5x 1 4. log6 36 2. 10log 5x 1. log7 7 eln 5 = 5 b. eln 5 1 − 3y = 3 2 1 y=− base blogbx = x, x > 0 log3 √ 3=y 3y = √ 3 b. log3 √ 3 Evaluate each expression. Evaluate each logarithm. Exercises ln e7 = 7 a. ln e7 Example 2 because 5-2 25 1 =− . 25 1 Therefore, log5 − = -2 5 =5 y = -2 y 1 a. log5 − 25 1 log5 − =y 25 1 y 5 =− 25 Example 1 The following properties are also useful. logb 1 = 0 logb b = 1 logb bx = x base by = x logb x = y if and only if Exponential Form Logarithmic Form If b > 0, b ≠ 1, and x > 0, then The inverse relationship between logarithmic functions and exponential functions can be used to evaluate logarithmic expressions. PERIOD 10/1/09 10:45:06 AM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 3 Logarithmic Functions and Expressions 3-2 NAME DATE (continued) PERIOD -2 0.028 -1 0.17 0 1 1 6 2 36 4 x→∞ 0 4 8 12 x Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 11 Chapter 3 −2.0 −1.0 0 1.0 2.0 y 2 4 1. g(x) = log3 x 6 x 11 Lesson 3-2 10/1/09 1:46:02 PM Glencoe Precalculus = -∞; decreasing (2, ∞) x → 2+ increasing (0, ∞) 7x lim 5 x → ∞ f(x) 3 D = (2, ∞); R = (-∞, ∞); y-intercept: none; x-intercept: (3, 0); asymptote: x = 2; end behavior: lim f(x) = ∞ and lim x → ∞ f(x) = ∞; x→0 −1.0 0 1 1.0 2.0 y 2. h(x) = -log3 (x - 2) D = (0, ∞); R = (-∞, ∞); y-intercept: none; x-intercept: (1, 0); asymptote: y-axis; end behavior: lim f(x) = -∞ and + Sketch and analyze the graph of each function below. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. Exercises Increasing: (0, ∞) x → 0+ End behavior: lim f(x) = -∞ and lim f(x) = ∞ Asymptote: y-axis Since the functions are inverses, you can obtain the graph of f(x) by plotting the points (f -1(x), x). y Domain: (0, ∞) 12 Range: (-∞, ∞) 8 x-intercept: (1, 0) f -1(x) x Create a table of values for the inverse of the function, the exponential function f -1(x) = 6x. Example Sketch and analyze the graph of f(x) = log6 x. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. The inverse of f(x) = bx is called the logarithmic function with base b, or f(x) = logb x, and read f of x equals the log base b of x. Logarithmic Functions Study Guide and Intervention Graphs of Logarithmic Functions 3-2 NAME Answers (Lesson 3-2) Chapter 3 DATE 2 9. log12 144 3 5 − 6. log8 32 4 3. log8 4096 PERIOD A5 0 y P0 12 Glencoe Precalculus 10/23/09 6:40:38 PM Answers Glencoe Precalculus x Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 12 Chapter 3 investment. An investment of $4000 was made in 2005 and had a value of $7500 in 2010. What was the average growth rate of the investment? 12.6% t 1 P ln − , where t is time in years, P is the present value, and P0 is the original r=− g(x) f(x) 1 . down and compressed horizontally by a factor of − 2 13. INVESTMENTS The annual growth rate r for an investment can be found using The graph of g(x) is the graph of f(x) shifted 2 units (2) x f(x) = ln x, g(x) = ln − -2 12. Use the graph of f to describe the transformation that results in the graph of g. Then sketch the graphs of f and g. decreasing: (-∞, ∞) decreasing: (-∞, ∞) x→∞ R = (0, ∞); y-intercept: (0, e) or (0, 2.72); x-intercept: none; asymptote: x-axis; lim f(x) = ∞ end behavior: x → -∞ lim and f(x) = 0; 8x D = (-∞, ∞); 4 R = (0, ∞); y-intercept: (0, 16); x-intercept: none; asymptote: x-axis; lim f(x) = ∞ end behavior: x → -∞ lim and x → ∞ f(x) = 0; 8x −4 −4 0 y D = (-∞, ∞); 4 −8 4 11. g(x) = e −8 −4 0 y 2x + 1 −6 −8 6 12 18 10. g(x) = 4 -x + 2 Sketch and analyze the graph of each function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. 0.014x 8. eln 0.014x 7. log6 216 3 18 5. 9 log 9 18 -3 10 4. 2 ln e5 3 1. log7 73 2. log10 0.001 Logarithmic Functions Practice Evaluate each expression. 3-2 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. P0 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 13 Chapter 3 answer: The population could not hit zero because there is an asymptote at y = 0. c. Will the population of the city ever reach zero if the trend continues? How do you know? Sample b. What is the population in 2010 and 2025? 50,000 and 49,729 a. Is the population growing or shrinking? shrinking 3. POPULATION The population P of a certain city in any year t since 2010 is given by the formula P = -100 ln t + 50,000. growth rate, t is time in years, P is the present value, and P0 is the original investment. If the original investment was $2500 in 2002 at a rate of 6.5%, in what year could you withdraw $10,000? 2023 t 1 P ln − , where r is the annual r=− 2. INVESTING The annual growth rate for an investment can be found using about 10,000 times as much c. How much more energy is released in an earthquake of magnitude 8.0 than of magnitude 3.0? 1000 kilowatt hours b. If an earthquake has magnitude 3.0, use your answer to part a to predict the amount of energy released. a. If the amount of energy released is 1 million kilowatt hours, what is the reading on the scale? 5.19 13 Logarithmic Functions DATE PERIOD 5 10 15 30 k Lesson 3-2 10/1/09 1:47:17 PM Glencoe Precalculus No; they need an interest rate of 13.9%. that earns 4.5% interest compounded continuously, will the investment double in 5 years? If not, what interest rate do they need? parents invested $5000 in an account rate written as a decimal. If Tara’s 5. FINANCIAL LITERACY The doubling time t, or the amount of time it takes for an investment to double, is given ln 2 by t = − , where k is the interest Sample answer: The rate of change is initially high, but then it slows down. d. What can you conclude about the rate of change in logarithmic functions? c. On which day is the population four times as high? 31 b. If the initial population is 25 bacteria, on which day will the population triple? 10 p 3 1 25 40 49 60 75 84 99 t 2 a. Complete the table below. Round to the nearest whole number. 4. BIOLOGY A certain strain of bacteria has a population that can be approximated by the formula P = 50 log 3.2t, where P is the population and t is the time elapsed in days. Word Problem Practice 1. SEISMOLOGY The Richter scale measures the magnitude M of an earthquake M = 0.67 log (0.37 E) + 1.46, where E is the energy released by the earthquake in kilowatt hours. 3-2 NAME Answers (Lesson 3-2) Enrichment PERIOD 9 7 10% 4 20% 3 30% 9 36 18 12 D2(r) 7 10% 4 20% 2 30% A6 Glencoe Precalculus 005_032_PCCRMC03_893804.indd 14 Chapter 3 14 504; No; sample answer: it would take about 252 years for your account to contain about $1,000,000. Glencoe Precalculus 5. Suppose your grandparents had placed $8000 dollars in an account when you were born. If your account earns r% compounded annually, what number would you divide by r to determine the approximate age at which the account will be worth about $1,000,000? If your account is earning 2%, would it be reasonable to assume that you would ever be a millionaire in your lifetime? Why or why not? 13 years; 5 years; 2 years; The estimates are very close to the calculated values. 4. Bankers have long used a rule known as the Rule of 72 to obtain a quick estimation of the number of years it takes to double an investment when compounded annually. They simply divide 72 by the interest rate to obtain a fairly accurate answer. How many years are needed to double an investment if it is earning 5.5%? 15%? 40%? How do the estimates using the Rule of 72 compare to the times calculated using the formula? The answers found in the table in Exercise 1 are very similar to the answers found in the table in Exercise 2. 3. Compare the results from Exercises 1 and 2. What do you observe? 8% 6% 4% 2% r 72 2. Now complete the table below by evaluating the function D2(r) = − r. 8% 6% D1(r) 4% 2% 35 18 12 r 1. Complete the table below. Find the number of years needed to double your money at each of the given interest rates. Round your answers to the nearest year. ln (1 + r) ln 2 , where r is written as a decimal. given by D1(r) = − of money invested and compounded annually at an interest rate of r% is The function D1 that gives the number of years needed to double the amount DATE 10/1/09 1:48:30 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 3 Double Your Money 3-2 NAME DATE PERIOD 3y 8x Expand ln − . 2 5 3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 15 Chapter 3 (x − 3)11 log9 − 32x5 4. 11 log9 (x − 3) − 5 log9 2x Condense each expression. 2 log3 5 + 5 log3 r − − log3 t 5r 2. log3 − 3 √t2 5 Expand each expression. 1 1. Evaluate 2 log3 27 + 4 log3 − . 3 2 = ln 8 + 5 ln x − ln 3 − 2 ln y 15 4 5 5 Lesson 3-3 3/18/09 6:01:11 PM Glencoe Precalculus ln ( √ (2h − k)3 √ (2h + k)3 ) 4 3 3 5. − ln (2h − k) + − ln (2h + k) log (a - 2) + 6 log (b + 4) – log 9 – 5 log (b - 2) (a − 2)(b + 4)6 9(b − 2) 3. log − 5 Power Property Product Property Quotient Property Simplify. =4 3y logx x = 1 = 3(3)(1) + 5(-1)(1) = ln 8 + ln x5 − ln 3 − ln y2 Exercises 2 1 8 = 2 3; 2 -1 = − Power Property 2 = 3(3 log2 2) + 5(-log2 2) 8x = ln 8x5 − ln 3y2 ln − 2 5 Example 2 2 1 Evaluate 3 log2 8 + 5 log2 − . Power Property Quotient Property Product Property 1 = 3 log 2 23 + 5 log 2 2−1 3 log2 8 + 5 log 2 − Example 1 • logb −xy = logb x − logb y • logb xp = p logb x • logb xy = logb x + logb y If b, x, and y are positive real numbers, b ≠ 1, and p is a real number, then the following statements are true. Since logarithms and exponents have an inverse relationship, they have certain properties that can be used to make them easier to simplify and solve. Properties of Logarithms Study Guide and Intervention Properties of Logarithms 3-3 NAME Answers (Lesson 3-2 and Lesson 3-3) Chapter 3 DATE Properties of Logarithms Study Guide and Intervention (continued) PERIOD A7 Use a calculator. Change of Base Formula Evaluate each logarithm. 8. log 11 47 1.606 7. log 6 832 3.753 16 Use a calculator. Change of Base Formula Glencoe Precalculus 10/1/09 10:48:42 AM Answers Glencoe Precalculus 12. log0.5 420 -8.714 9. log 3 9 2 6. log 9 712 2.99 3. log 7 1094 3.60 ≈ -2.10 3 1 log − Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 16 Chapter 3 11. log 12 4302 3.367 5. log 5 256 3.45 4. log 6 94 2.54 10. log8 256 2.667 2. log 3 17 2.58 3 log 10 log −1 10 = − 3 b. log −1 10 ln b ln x log b x = − 1. log 32 631 1.86 Evaluate each logarithm. Exercises ≈ 2.81 ln 7 log 2 7 = − ln 2 a. log2 7 Example log b x = − log x log b Many non-graphing calculators cannot be used for logarithms that are not base e or base 10. Therefore, you will often use this formula, especially for scientific applications. Either of the following forms will provide the correct answer. log a x For any positive real numbers a, b, and x, a ≠ 1, b ≠ 1, log b x = − . log a b If the logarithm is in a base that needs to be changed to a different base, the Change of Base Formula is required. Change of Base Formula 3-3 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Properties of Logarithms Practice DATE 4 3 2 (x + 1) 3 √x +5 10. log 2 − 3 3 3 log 8 (4x + 2) + log 8 (x − 4) 8. log8 [(4x + 2) 3(x − 4)] 2 1 2 ln x - − ln (3x + 2) √ 3x + 2 x 6. ln − 3x − 5y √ 3 2 5 I0 1.774 21. log 6 24 1.151 18. log 100 200 2 log 2 − (x + 4) √x log 3 − xyz 3 2 3 −2.078 22. log −1 9.8 -0.301 19. log 0.01 4 1 16. log 3 y + log 3 x – − log 3 x + 3 log 3 z 8x log 3 − 2 √ x−4 14. log 3 8 + log 3 x – 2 log 3 (x + 4) (5x + 6) 3 1 12. 3 log 2 (5x + 6) − − log2 (x − 4) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 17 Chapter 3 17 b. Macquarie Island region, in 2008, R = 7.1 about 12,589,254 a. Guam region, in 2008, R = 6.7 about 5,011,872 I 0 = 1. Find the intensity per unit of area for the following earthquakes. Lesson 3-3 10/1/09 1:51:14 PM Glencoe Precalculus 23. SEISMOLOGY The intensity of a shock wave from an earthquake is given by the I , where R is the magnitude, I is a measure of wave energy, and formula R = log 10 − -4.046 20. log 0.24 322 2.322 1 17. log −1 − Evaluate each logarithm. √x log − 3 3yz 2 1 15. log y + log 3 − − log(x) + 2 log z 6x 49 log 7 − 2 13. 2 - log 7 6 – 2 log 7 x 4x + y ln − 2 1 11. − ln (3x – 5y) – ln (4x + y) Condense each expression. 1 1 log 13 3 + 4 log 13 x − − log 13 (7x − 3) 3 log 2 (x + 1) − − log 2 (x + 5) 9. log 13 3x 4 − 3 √ 7x − 3 3 + 3 log 2 x + log 2 (x + 1) 7. log2 [(2x) 3(x + 1)] 1 ln (x + 1) − − ln (x − 5) 5. ln − 4 x+1 √ x−5 Expand each expression. 10000 729 4. ln − 6 ln 3 − 4 ln 10 ln 10 − 2 ln 3 2. ln 27000 3 ln 3 + 3 ln 10 10 3. ln − 9 PERIOD 1. ln 300 ln 3 + 2 ln 10 Express each logarithm in terms of ln 10 and ln 3. 3-3 NAME Answers (Lesson 3-3) A8 Glencoe Precalculus 005_032_PCCRMC03_893804.indd 18 Chapter 3 ( ) r log E = n log 1 + − n -1 4. The effective annual yield E for an account compounded n times per year at n percent is modeled by the equation r n log E = log (1 + − n ) - 1. Rewrite the equation using the properties of logarithms. 6.23 characters, or 7 characters will make 7.2 × 10 16 tries necessary 3. TECHNOLOGY Computer hackers are often very interested in passwords to gain access to systems. There are about 256 keys someone can use to make a password character. In order to guess a password of length L, a hacker would have to try N different combinations to access the system. If the formula is L = log 256 N, how long do you have to make your password in order to guarantee at least 10 15 tries by hackers? 2. BIOLOGY A formula to express the amount of time it takes to double a certain colony of bacteria is rt = ln 2, where r is the growth rate and t is the doubling time in hours. How long will it take to double a bacteria colony if it grows at a rate of 5% per hour? 13.9 h 18 Properties of Logarithms ( PERIOD P ) Glencoe Precalculus d. Since 2004, the likelihood of that impact has dropped to 1 in 45,000. What is the new Palermo Scale number for 99942 Apophis? -2.0 c. The threshold for monitoring an object with a potential to collide with Earth is a Palermo Scale number between −2 and 0. There is some concern if an object has a number higher than 0. In 2004, the object 99942 Apophis broke the record for the highest Palermo Scale number. It had a 2.7% chance of collision with Earth in 2029 with an energy of 26,000 Megatons. How high was the Palermo Scale number? 1.1 b. What is the Palermo Scale number for an asteroid which has a 12% chance to strike the Earth in 20 years with an energy of 0.45 Megatons? -1.98 5 P = log P i − log 0.3 + 4 − log E − log T a. Expand the formula using the properties of logarithms. P is the Palermo Scale number, P i is the probability of impact estimated by the path of the object, E is the energy in Megatons of TNT of the impact, and T is the time in years away of the estimated impact. 0.3E 5 T − i occurrence is P = log − , where −4 5. ASTRONOMY The Palermo Technical Impact Hazard Scale is a logarithmic scale used by astronomers to rate the potential danger from asteroids colliding with the Earth and causing a large amount of damage. The formula to measure the probability of such an Word Problem Practice DATE 3/18/09 6:01:25 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 3 1. CHEMISTRY A useful way to express the half-life of a radioactive object is in terms of the time it will take to decay to a percentage P of its original value given by the formula t = −T log 2 P, where T is the half-life and t is the time elapsed to the percentage in years. How long will it take Carbon-14 to decay to 20% of its original amount if the half-life is 5730 years? 13,305 yr 3-3 NAME Enrichment DATE PERIOD ⎪x ⎥ ) 1 + √ 1 - x2 x+1 x-1 sech-1 x = ln − ; (0, 1] x 2 1 coth-1 x = − ln − ; (-∞, -1) ∪ (1, ∞) cosh-1 x = ln (x + √ x 2 - 1 ); [1, ∞) ex + e-x 2 ) -2x Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 19 Chapter 3 19 Lesson 3-3 3/18/09 6:01:29 PM Glencoe Precalculus Show that the f(x) = cosh x and g(x) = cosh-1 x are inverses. See students’ work. Exercise 2 -x 2 ) =x -x =x 2 = ln (ex) x 2 2x =− ( x e -e = ln − +− x + √x +1 − 2 2 2x (x + √ x2 + 1 ) =− 2 2 1 ln − 2x -x 2 e -2+e ) + −44 ) (− √ 4 x e -e − + 1) √( 2 ) (e + e ) √ ex - e-x = ln − +− 2 2 2 2 1 x + √ x2 + 1 - − x + √ x2 + 1 = −− 2 ( ( ( ex - e-x g(f(x)) = ln − + ex - e-x = ln − + 2 ln (x + √ x +1) x +1 ) - e ( x+ √ = e−− 2 ln (x + √ x +1) x +1) - e -ln (x + √ f(g(x)) = e−− 2 tanh x 1 coth x = − ,x≠0 Show that f(x) = sinh x and g(x) = sinh-1 x are inverses. Show that f(g(x)) = g(f(x)) = x. Example √ 1 + x2 1 csch-1 x = ln − x + − ; (-∞, 0) ∪ (0, ∞) ( 1 tanh-1 x = − ln − ; (-1, 1) 1+x 1-x x 2 + 1 ); (-∞, ∞) sinh-1 x = ln (x + √ 2 cosh x sinh x tanh x = − Definitions of the Inverse Hyperbolic Functions 1 sech x = − cosh x 1 ,x≠0 csch x = − sinh x 2 Definitions of the Hyperbolic Functions x + e-x cosh x = e− 2 -x -e sinh x = e− x Vincenzo Riccati introduced the hyperbolic functions in the mid-18th century. They were first used to compare the area of a semicircular region with the area of a region under a hyperbola. These functions have properties similar to the trigonometric functions you will study in Chapters 4 and 5. The definitions of the hyperbolic functions and their inverses are given below. Hyperbolic Functions and Their Inverses 3-3 NAME Answers (Lesson 3-3) Chapter 3 DATE Exponential and Logarithmic Equations Study Guide and Intervention PERIOD A9 x ≈ 2.68 log 19 x=− log 3 e2x - 3ex + 2 = 0 u2 - 3u + 2 = 0 (u - 2)(u - 1) = 0 u = 2 or u = 1 ex = 2 or ex = 1 x = ln 2 or 0 8 5 Glencoe Precalculus 10/1/09 1:52:47 PM Answers Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 20 20 8. e 19x = 23 0.17 7. 32x = 6x − 1 −4.42 Chapter 3 calculator. Divide by 8 and use a Subtract 1 from each side. ln e = 1 Power Property Take the ln of both sides. Add 6 to both sides. 4. 2e2x + 12ex - 54 = 0 1.10 (8) 1 = − 11 − x 6. 2.4ex − 6 = 9.3 7.35 2 1 − (4) 1 2. − 2x − 1 5. 92x = 12 0.57 3. 43x - 2 1 2x =− 2 1. 9x = 33x - 4 4 logarithm of each side. Take the natural Substitute for u. Solve. Factor. Write in quadratic form. Original equation b. e8x + 1 - 6 = 1 e8x + 1 = 7 Take the log of both sides. ln e8x + 1 = ln 7 Power Property (8x + 1) ln e = ln 7 Divide each side by log 3. 8x + 1 = ln 7 Use a calculator. Check this 8x = ln 7 − 1 solution in the original equation. ln 7 − 1 x=− ≈ 0.12 Solve each equation. Round to the nearest hundredth. Exercises Subtract x from each side. One-to-One Property Power of a Power 16 = 4 2 Original equation b. Solve e2x − 3ex + 2 = 0. Solve each equation. Round to the nearest hundredth if necessary. log 3x = log 19 x log 3 = log 19 a. 3x = 19 Example 2 4x - 1 = 16x 4x - 1 = (42)x 4x - 1 = 42x x - 1 = 2x -1 = x a. Solve 4x − 1 = 16x. Example 1 This property will help you solve exponential equations. For example, you can express both sides of the equation as an exponent with the same base. Then use the property to set the exponents equal to each other and solve. If the bases are not the same, you can exponentiate each side of an equation and use logarithms to solve the equation. One-to-One Property of Exponential Functions: For b > 0 and b ≠ 1, bx = by if and only if x = y. Solve Exponential Equations 3-4 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. DATE PERIOD (continued) Exponential and Logarithmic Equations Study Guide and Intervention Factor. Solve. 7 12 4. log3 3x = log3 36 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 21 Chapter 3 6. ln x + ln (x + 16) = ln 8 + ln (x + 6) 4 21 5. log (16x + 2) + log (20x - 2) = log (319x2 + 9x - 2) 2 no solution 3. log (x + 1) + log (x - 3) = log (6x2 - 6) 4 1. log 3x = log 12 yields logs of negative numbers. Therefore, -2 is extraneous. x = 8 log2 (8 − 6) = 5 − log2 [2(8)] log2 2 = 5 − log2 16, which is true. Therefore, x = 8. x = −2 log2 (−2 − 6) = 5 − log2 [2(-2)] CHECK Lesson 3-4 3/18/09 6:01:38 PM Glencoe Precalculus 2. log12 2 + log12 x = log12 (x + 7) exponential form. Expand. Solve each logarithmic equation. Exercises 2x2 − 12x − 32 = 0 2(x - 8)(x + 2) = 0 x = 8 or -2 Rewrite in Product Property Rearrange the logs. Original equation Solve log2(x − 6) = 5 - log2 2x. Use a calculator. Divide each side by 4. Write in exponential form. (Use 5 as the base when exponentiating.) log2 (x - 6) = 5 - log2 2x log2 (x - 6) + log2 2x = 5 log2 (2x(x - 6)) = 5 2x(x - 6) = 2 5 Example 2 x = 3906.25 4 Divide each side by 2. Add 1 to each side. Original equation Solve 2 log5 4x − 1 = 11. 2 log5 4x − 1 = 11 2 log5 4x = 12 log5 4x = 6 4x = 5 6 56 x=− Example 1 This property will help you solve logarithmic equations. For example, you can express both sides of the equation as a logarithm with the same base. Then convert both sides to exponential form, set the exponents equal to each other and solve. One-to-One Property of Exponential Functions For b > 0 and b ≠ 1, logbx = logby if and only if x = y. Solve Logarithmic Equations 3-4 NAME Answers (Lesson 3-4) 6 5 A10 2 16. log5 (x + 4) + log5 x = log5 12 2 no solution 14. log (2x + 1) + log (x − 4) = log (2x2 - 4) 20. 4 = 1056 1.67 23. 6 2x - 1 = 18 1.31 19. 2 = 1210 1.14 22. 3 x − 8 + 2 = 38 11.26 - 4 = 19 −1.05 Glencoe Precalculus 005_032_PCCRMC03_893804.indd 22 Chapter 3 22 26. FINANCIAL LITERACY If $500 is deposited in a savings account providing an annual interest rate of 5.6% compounded quarterly, how long will it take for the account to be worth $750? 7.29 years Glencoe Precalculus 24. 2 3 + 2x = 130 2.01 21. 5 x+3 25. BANKING Ms. Cubbatz invested a sum of money in a certificate of deposit that earns 8% interest compounded continuously. The formula for calculating interest that is compounded continuously is A = Pe rt. If Ms. Cubbatz made the investment on January 1, 2005, and the account was worth $12,000 on January 1, 2009, what was the original amount in the account? $8713.79 9x 3x Solve each equation. Round to the nearest hundredth. no solution 17. log (x + 1) + log (x − 3) = log (6x - 6) 18. ln 0.04x = -8 ≈0.008 15. 2 -4x + 1 = 3 2x - 3 ≈0.80 0 and ≈0.811 13. 4e 2x - 13e x + 9 = 0 ≈0.28 and ≈-0.23 12. 6 ln (x + 2) - 3 = 21 ≈52.6 ≈-1.639 11. 6e6x - 17e 3x + 7 = 0 =6 -2x - 3 10. 4 x+2 40 9. ln (x - 5) + ln 4 = ln x - ln 2 − 7 8. 3e4x - 9e2x - 15 = 0 ≈0.7167 17 2 4. e2x - ex - 6 = 0 ≈1.099 7. ln (2x - 1) = ln 16 − = 27x - − 1 6. ln − e = x -1 x+3 5. logx 64 = 3 4 1 3. − (9) 1. 5x = 125 x-2 3 PERIOD 2. log6 x + log6 9 = log6 54 6 Exponential and Logarithmic Equations Practice DATE 10/1/09 1:55:55 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 3 Solve each equation. 3-4 NAME 1000 1250 1563 1 2 3 4. INTEREST RATE The effective annual yield E for an account that is compounded n times per year at r percent is given by the formula r n E = 1 + − n - 1. Suppose an account pays 5%. Use a calculator to find how many compounding periods it would take for the effective yield to be 5.1%. about 17,288,942 kilowatt hr 3. RICHTER SCALE The function used to measure the magnitude R of an earthquake is given by R = 0.67 log (0.37E) + 1.46, where E is the energy in kilowatt hours that is released by the earthquake. If the magnitude of an earthquake is 6.0, find the approximate energy released. b. In what week will the blog get 10,000 hits? 12th week N = 1000(1.25)w - 1 a. Write an exponential model to find the number of hits N in week w. Number of Hits Week 2. INTERNET The table shows the number of hits an Internet blog received for three weeks. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 23 Chapter 3 5 DATE 23 PERIOD 2 3 24 4 32 5 40 −t 8 t-2 − Lesson 3-4 10/1/09 1:56:46 PM Glencoe Precalculus d. Using the formula in part c, find how long it will take for the number of bacteria to reach 5000. 39.2 hr N = 200(2) c. Suppose a chemical is added that kills exactly one half of the bacteria at the 16-hour mark. Write an equation to find the number of bacteria after the 16-hour mark. N = N0(2) 8 , or N = 100(2) 8 −t 20040080016003200 16 8 1 b. Write an equation to model the number of bacteria N at time t given N0, the initial number of bacteria. Population Time (hr) Number of doubles a. If there were 100 bacteria at t = 0 hours, complete the table below. 6. BIOLOGY A certain strain of bacteria in a perfect growth medium doubles in population every 8 hours. 5. RADIOACTIVITY The amount of radioactivity in a sample is given by the equation ln (N) - ln (N0 ) = −kt, where N is the current level, N0 is the original level, k is the decay rate, and t is the time elapsed in hours. If the decay rate is 0.070, how many grams would be left after 24 hours if the original amount was 1000 grams? 186.4 grams Exponential and Logarithmic Equations Word Problem Practice 1. RADIOACTIVE DECAY The amount of radium A present in a sample after t years can be modeled by A = A0e -0.00043t, where A0 is the initial amount. How long will it take 50 grams to decay to 10 grams? 3743 years 3-4 NAME Answers (Lesson 3-4) Chapter 3 Enrichment DATE PERIOD A11 1.36 years 24 Glencoe Precalculus 10/1/09 1:57:29 PM Answers Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 24 Chapter 3 7. Suppose the computer’s value suddenly drops because of a surplus at the end of the third year. The company can sell the used computer for one quarter as much as it could before the surplus. What is the new residual value of the computer? At what time does the book value match the residual value? $135; 5.86 years exponential model d. In which year is the depreciation expense zero? cannot be zero with the c. Calculate the depreciation expense for the second year. $600 b. If the company takes in $10,000 in the first year, what should the earnings report show as a balance? $9000 a. Calculate the depreciation expense for the first year. $1000 6. An earnings report shows the amount of money that the company takes in minus (among other things) the depreciation expense. over in year 3. 5. The company has decided to buy new computers every five years. Is this a wise strategy based on this estimated useful life? Explain. No; the useful life of the machine is 4. What is the estimated useful life of the machine (in whole years)? 3 years 3.15 years 3. How long will it take for the book value to be the same as the residual value of $500? 2. How long will it take for the book value be one-half of the asset cost? V = 2500(0.60)t 1. Write a formula to give the book value V of the computer t years after purchase. A computer is purchased for $2500. Its depreciation rate is 40% per year. The book value of an asset is equal to the asset cost minus the accumulated depreciation. This value represents the unused amount of asset cost that the company may depreciate in future years. There are several methods of determining the amount of depreciation in a given year. In the declining-balance method, the depreciation expense allowed each year is equal to the book value of the asset at the beginning of the year times the depreciation rate. • residual value: the expected cash value of the asset at the end of its useful life • estimated useful life: the number of years the company can expect to use the asset • asset cost: the amount the company paid for the asset For tax purposes, a company distributes the cost of assets, such as equipment or a building, as an expense over the course of a number of years. Because assets depreciate (lose some of their value) as they get older, companies are allowed to deduct depreciation when filing income taxes. Depreciation expense is a function of these three values. Depreciation 3-4 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Graphing Calculator Activity DATE PERIOD [0, 100] scI: 10 by [0, 5] scI: 1 2. log2 (x - 1) ≤ 2 (1, 5] 3. 0 ≤ log5 x ≤ 1 (0, 5] Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 25 Chapter 3 25 will an investment double in less than 5 years? r ≥ 14.87% ln (1 + r) ln 2 value at a rate of r percent is given by t = − . For what values of r Lesson 3-4 10/1/09 2:37:41 PM Glencoe Precalculus 5. INVESTMENT The length of time t it takes for an investment to double in 4. DEPRECIATION A family buys a car for $25,000. If it depreciates at a rate of 10% per year, for how long will the value of the car be above $15,000? up to 4.85 years 1. log3 x ≥ 2 [9, ∞) Solve each exponential and logarithmic inequality using a table or a graph. Exercises Y2 = 3 Use the intersect feature to find the point at which the logarithmic function is equal to 3. The graph is below the line y = 3 when x < 64. So, log4 x ≤ 3 for (0, 64]. Y1 = − log (X) log (4) Example 2 Solve log4 x ≤ 3. Use a graph. Use the change of base formula to enter f(x) = log4 x into a graphing calculator. Also, enter f(x) = 3. Use a table. Enter the function f(x) = 150,000(1.03)x into a graphing calculator. Use [Tblset] to make the increment 0.1 year, and use the table feature to show the number of years and the value of the home. So, it takes about 5.3 years for the home to be worth $175,000. Example 1 Suppose your family buys a home for $150,000. Its value increases by 3% per year. When will the value of the home exceed $175,000? Let x represent the time in years. Then a function relating x and the value of the home is f(x) = 150,000(1.03)x. To find when the value of the home will exceed $175,000, solve the inequality 150,000(1.03)x > 175,000. Tables, graphs, and graphing calculators can be used to help solve exponential and logarithmic inequalities. Solve Exponential and Logarithmic Inequalities 3-4 NAME Answers (Lesson 3-4) Modeling with Nonlinear Regression Study Guide and Intervention DATE 1777.30 GDP 2501.80 1960 3771.90 1970 5161.70 1980 7112.50 1990 9817.00 2000 A12 Glencoe Precalculus 005_032_PCCRMC03_893804.indd 26 Chapter 3 26 Sample answer: The number of mice being born with the defect has stabilized. 4. Make a conjecture about why a logistic model may fit the data. 9.639 Glencoe Precalculus 7 9.019 7.489 4 6 6.618 3 8.297 5.721 2 5 y 4.836 x 1 [0, 70] scI: 10 by [1000, 10,000] scI: 1000 [0, 10] scI: 1 by [0, 15] scI: 1 3. Find the value of the model at x = 20. f(20) = 11.94 12 f(x) = − 1 + 2e-0.3x 2. Find a logistic function to model the data. 1. Make a scatter plot of the data. The data shows the number of mice per thousand who were born with a certain genetic characteristic over the last 7 years. Exercises Step 1 Make a scatter plot. Step 2 Find a function to model the data. The graph appears to be rapidly rising. Try an exponential function. With the diagnostic feature turned on and using ExpReg from the list of regression models yields the values shown in the [0, 70] scI: 10 by [1000, 10,000] scI: 1000 second screen. The GDP in 1950 is represented by a, and the growth rate, 3.5% per year, is represented by 1 - b. Notice that the correlation coefficient r ≈ 0.999 is close to 1, indicating a close fit to the data. Step 3 Graph the formula and the scatter plot on the same screen. In the Y= menu, pick up this regression equation by entering VARS , Statistics, EQ. The exponential fit is nearly perfect. Step 4 Estimate the 2020 GDP. Depending on the fit, it should be around 19,948. Source: U.S. Dept. of Commerce 1950 Year Example Use the data in the table to determine a regression equation that best relates the gross domestic product, GDP, (in billions of chained 2000 dollars) to the year. Estimate the 2020 GDP. Regression can be used to model real-world data that exhibits exponential or logarithmic growth or decay. PERIOD 10/1/09 10:58:51 AM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 3 Exponential, Logarithmic, and Logistic Modeling 3-5 NAME DATE Modeling with Nonlinear Regression Study Guide and Intervention (continued) PERIOD 2 765 4 916 6 8 10 12 14 1004 1066 1115 1155 1188 765 Machines 916 1.386 1004 1.792 1066 2.079 1115 2.303 1155 2.485 1188 2.639 Brightness (candelas) 30 8.16 40 4.59 50 60 2.04 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 27 27 Sample answer: exponential, f(x) = 33.15(0.96)x; brightness at 100 = 0.56 candelas Chapter 3 70 1.15 80 [0, 4] scI: 1 by [600, 1300] scI: 75 [0, 15] scI: 2 by [600, 1300] scI: 75 Lesson 3-5 10/1/09 2:37:51 PM Glencoe Precalculus [14, 86] sci: 1 by [-0.33, 23] sci: 5 1.50 [14, 86] scI: 1 by [-0.33, 23] scI: 1 2.94 c. Calculate an exponential or logarithmic regression model for the data and estimate the brightness at 100 meters. Sample answer: graph (x, ln y) for each point in the table. b. Create a graph that linearizes the data. exponential a. Make a scatter plot of the data and describe the possible model. 20 18.36 Distance (m) The brightness of a point of light is measured at different distances. Exercise Make a scatter plot of this data and draw a line through it. This creates a straight line. Therefore, the original function is logarithmic. y = 217x + 614 y = 217 ln x + 614 On day 9, there are y = 217 ln (9) + 614 = 1091 infected machines. 0.693 ln (day) Step 1 Make a scatter plot of this data to get a sense of the shape. The shape of the graph suggests a logarithmic function. Step 2 Create a graph for (ln x, y). Machines Day Example The spread of a given computer virus is measured over time. What model will best describe its growth? How many machines would be infected on day 9? • a power function y = axb, graph (ln x, ln y). • a logarithmic function y = a ln x + b, graph (ln x, y). • an exponential function y = abx, graph (x, ln y). • a quadratic function y = ax2 + bx + c, graph (x, √ y ). The correlation coefficient is a measure calculated from a linear regression. Data can be linearized in order to measure a correlation coefficient for nonlinear data. This is accomplished by graphing (x, r(x)), where r(x) is the regression model you are using, such as power, exponential, or logarithmic. If the data appears to be in a cluster about a line, then this regression model is likely to be a good fit. To linearize data modeled by: Linearizing Data 3-5 NAME Answers (Lesson 3-5) Chapter 3 DATE 5.1 y 11.4 34 25.6 36 57.7 38 129.7 40 291.9 42 32.4 656.8 1477.9 46 A13 [0, 50] scI: 10 by [0, 10] scI: 1 [30, 48] scI: 4 by [0, 1500] scI: 175 y 20 3.05 30 3.98 40 4.06 50 4.51 60 5.13 70 5.49 80 5.25 28 90 100 5.38 Glencoe Precalculus 10/1/09 2:38:03 PM Answers Glencoe Precalculus [0, 110] scI: 10 by [0, 2] scI: 0.2 [0, 110] scI: 10 by [0, 10] scI: 1 5.11 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 28 Chapter 3 1+e 5.443776 logistic; y = − -0.052x d. Use the linear model to find a model for the original data. Check its accuracy by graphing. c. Graph the linearization and write the equation of the line of best fit. y = 0.008x + 1.02 10 20 30 40 50 60 70 80 90 100 x ln y 0.88 1.12 1.38 1.40 1.51 1.64 1.70 1.66 1.63 1.68 b. Linearize the data according to the given model. a. Make a scatter plot of the data. 10 2.41 x 4. The data below is for the population of Chicago, Illinois, and Cook County, in millions, from 1910–2000, where x is the number of years since 1900. d. Use the linear model to find a model for the original data. x Check its accuracy by graphing. y = 0.0000119(1.5) c. Graph the linearization and write the equation of the line of best fit. y = 0.4051x - 11.340 32 34 36 38 40 42 44 46 x ln y 1.629 2.433 3.243 4.055 4.866 5.676 6.487 7.298 b. Linearize the data according to the given model. a. Make a scatter plot of the data. 32 3. x 19,678 c. Find the value of the model at x = 25. c. Find the value of the model at x = 20. [0, 12] scI: 2 by [0, 24] scI: 4 b. Find a logarithmic function to model the data. y = 10.8 ln (x)- 2.4 44 3 4 5 6 7 8 9 9.5 12.6 15.0 17.0 18.7 20.1 21.4 a. Make a scatter plot of the data. 2 5.1 PERIOD b. Find an exponential function to model the data. y = 0.333e0.5493x [0, 10] scI: 2 by [0, 30] scI: 5 a. Make a scatter plot of the data. y 2. x Modeling with Nonlinear Regression Practice 1 2 3 4 5 6 7 8 y 0.58 1.00 1.73 3.00 5.20 9.00 15.59 27.00 1. x 3-5 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 29 Chapter 3 Month Hits 2 82 4 6 8 10 12 170 265 369 480 601 2. INTERNET A popular blog has steadily grown in the number of times the main page is read (hits) each month. Advertisers pay more for more popular sites, so an accurate calculation of projected traffic is critical. Use a scatter plot and an exponential model to find the number of hits at the end of the second year. around 7000 hits around 11.13 billion c. Use the type of model to predict the population of the world in 2050. exponential b. From the graph, what seems to be the most appropriate model for the data? Sample answer: [0, 100] scI: 10 by [0, 6] scI: 1 a. Make a scatter plot of the data. Source: United Nations PERIOD 1 + 9.17e 4. BACTERIA Bacteria that is cultured in a petri dish cannot continue to grow forever in an exponential manner because resources become limited. The table below estimates the population of bacteria in a culture, in thousands. 2007, there would have been about 227,000 rats using this model. That would be about 1 rat per 36.6 people; the estimate is very close. b. A better estimate is that there is a maximum of about 1 rat per 36 people. The estimated population of New York in 2007 was about 8.3 million. How close is your model’s estimate? Sample answer: In Sample answer: 230 y= − -0.1t a. Find a logistic model for this data. Year 1940 1950 1960 1970 1980 1990 2000 Rats 22.6 52.6 102.7 158.0 197.0 216.6 224.9 Lesson 3-5 10/1/09 2:38:17 PM Glencoe Precalculus the limiting resources would be removed and the function would not be logistic. c. If the culture was suddenly expanded in size and food was continuously added to the colony, would this model still hold? Why or why not? No, 3,124,000 b. Estimate the population of bacteria that will exist at hour 120. 1 + 270e 3129 y= − -0.1x Sample answer: a. Find a logistic model for the data. Hour 12 24 36 48 60 72 84 Population 29.2 121.1 390.3 1018.3 1932.8 2642.5 2970.3 29 1930 1940 1950 1960 1970 1980 1990 2000 Year Population 2.07 2.30 2.52 3.02 3.70 4.44 5.27 6.06 DATE 3. POPULATION A common statement is that there is one rat per person in New York City. Below is a fictional table with data regarding the number of rats (in thousands) in the city in a given year. Word Problem Practice 1. POPULATION The table below gives the population of the world in billions for selected years. 3-5 NAME Answers (Lesson 3-5) Enrichment PERIOD 10 20 30 40 50 60 70 80 90 100 3.5 3.7 3.9 4.2 4.7 5.1 5.4 A14 10 5.1 325,691 5.5 5.3 30 210,551 20 5.6 398,564 40 5.8 614,799 50 Glencoe Precalculus 005_032_PCCRMC03_893804.indd 30 Chapter 3 30 Since the data lie in a relatively straight line after 1940, you can say that Dallas has experienced exponential growth from 1920 to 1990. Years since 60 70 80 90 100 1900 x Population y 951,527 1,327,321 1,556,390 1,852,810 2,218,899 log y 6.0 6.1 6.2 6.3 6.3 log y Population y 135,748 Years since 1900 x The table below shows the population of Dallas, Texas, and the surrounding county from 1910 to 2000. Determine if an exponential model is a good fit for the data. Exercise Step 3 Interpret the graph. Since the data lie in a relatively straight line after 1940, you can say that Las Vegas has experienced exponential growth since 1940. Step 2 Graph (x, log y) on the semilogarithmic paper. log y 1 2 3 4 5 6 10 9 8 7 1 2 3 4 5 6 10 9 8 7 5.7 20 20 40 40 5.9 60 60 Glencoe Precalculus Years 80 100 120 140 80 100 120 140 Years 6.3 Population y 3321 4859 8532 16,414 48,289 127,016 273,288 463,087 741,459 1,836333 Years since 1900 x Step 1 Fill in the table values. Round to the nearest tenth. Example The table below shows the population of Las Vegas, Nevada, and the surrounding county from 1910 to 2000. Determine if an exponential model is a good fit for the data. Regression equations for exponential functions may depend on creating a table of values with x as the independent variable and log y as the dependent variable, as shown in the example below. If you use semilogarithmic paper, you can graph the table of values in a simplified way. DATE 3/18/09 6:02:25 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 3 Graphing on Logarithmic Paper 3-5 NAME Spreadsheet Activity DATE PERIOD Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC03_893804.indd 31 Chapter 3 31 Yes, the NPV is $868.40 greater than the cost. b. Would this be a good investment of $3000? Explain. 3. a. Calculate the NPV for an investment over a period of six years if the cost of capital is 4.5% and the investment will bring a cash flow of $750 every year. The NPV would be about $3868.40. Sample answer: Yes. After one year, the gain is $3840.05, so they should buy the printer. a. The last issue of the magazine sold 500 copies. If each issue of the magazine printed in color sells 100 copies more than the previous issue, is the printer a good investment after one year? Explain. 7924.664 1707.534 1878.287 2066.116 2272.727 -8000 (1 + r) CF − n Lesson 3-5 3/18/09 6:02:30 PM Glencoe Precalculus 2. Four times a year, Josey and Drew publish a magazine. They want to buy a color printer that costs $1750. The cost of capital for this purchase would be 6%. They are planning to raise the price of their magazine from $1 to $2. Create a spreadsheet to determine the NPV for this purchase. greater than the NPV, so it would not be cost-effective to buy the van. 1. If the NPV is greater than the cost, the investment will pay for itself. Based on the spreadsheet shown above, would it be cost-effective for the company to buy the van? Explain. The cost is actually about $75 Exercises and r = the cost of capital, which is either the interest that will be paid on a loan or the interest that the money would earn if it were invested. (1 + r) n NPV = − , where CFn = the cash flow in period n n CF When a business owner is considering Period (n) CF a major purchase, it is important to determine whether the investment will be profitable in the 0 -8000 future. Consider the example of a local 1 2500 restaurant-delivery service that is debating whether 2 2500 to buy a van for $8000. The owners of the company estimate that using the van will generate $2500 3 2500 per year over four years. They can use the 4 2500 following formula to find the present value of the Total future cash flow to find the Net Present Value (NPV), NPV - Cost -75.3364 that is, how much the profits would be worth in Cost of Capital (r) 10% today’s dollars. Net Present Value 3-5 NAME Answers (Lesson 3-5) Chapter 3 Assessment Answer Key Quiz 1 (Lessons 3-1 and 3-2) Page 33 1. 81 D 16 1. A 2. G 3. D 4. J 5. A 9 3. 4. Page 35 B 2. 3. Mid-Chapter Test 7.3 yr 1 = -4 log 3 − 4. (Lesson 3-4) Page 34 1. $16,976.03 2. Quiz 3 0.46 y 2 5. 12 8 4 -2 0 5. 2 4 6x Quiz 4 (Lesson 3-5) y Page 34 x 1. Quiz 2 (Lesson 3-3) Page 33 1. 2.289 2. 0.8 3. 6 2. D ln (1 - p) + ln d 1 − ln (2 + 5r) 5. 6 Chapter 3 Sample answer: y = 0.24 (2.34) x Sample answer: 3. 215.7 6. ln (x + 1) 1 − ln (x - 5) 4 2x + y log3 − 3 4. 4. B Answers 0 Sample answer: exponential; y = 15.198 (1.07) x Sample answer: y = 0.0058x + 5. 3.726 7. x-y √ 8. 295 No; she will have only 9. $1760.27. 10. A15 17,239 Glencoe Precalculus Chapter 3 Assessment Answer Key Vocabulary Test Page 36 Form 1 Page 37 1. transcendental 1. function 2. Page 38 A D 12. H 13. D 14. F 15. C 16. G 17. A 18. G H 2. algebraic function 3. 11. exponential function 3. A logistic growth 4. function 5. common logarithm 6. natural logarithm 4. J 5. B 6. J 7. B 8. G 9. C 20. G 10. J B: D 7. linearize 8. natural base 19. A 9. an account with no waiting period between interest payments 10. the inverse of the function f(x) = b Chapter 3 x A16 Glencoe Precalculus Chapter 3 Assessment Answer Key Form 2A Page 39 1. 2. 3. B Page 40 11. D 12. Page 42 11. A H 12. F 13. A 13. A 14. F 14. J 15. B 15. B 16. H 16. H 17. A 18. H 19. B 20. J 7 log 7 2x H C Form 2B Page 41 1. 2. B J 3. C 4. F 4. F 5. C 5. B 6. H 6. H 7. D 8. F 9. B 10. J Chapter 3 18. J 19. A F 20. B: 11 log 11x 7. D 8. G 9. D 10. A17 G Answers 17. A B: Glencoe Precalculus Chapter 3 Assessment Answer Key Form 2C Page 43 Page 44 y 1. 8 exponential 12. 3.14 13. 1.15 x 0 2. 11. y 4 −8 4 −4 0 3 14. −4 −8 15. log913 + 2 log9 x - 5 log 9 y $1987.87 3. 2.49 g 4. 5. x2 log − 8x log −1 1296 = -4 6 3 -− 4 6. 13 17. 3 shifted 2 units right, expanded vertically by a factor of 3, and 18.reflected in the x-axis 5 7. 0 9. 2.776 10. -7.432 8.66% 19. y 8. Chapter 3 16. x In y = 0.0582x + 4.4657 20. 3 − = loga x B: A18 2 Glencoe Precalculus Chapter 3 Assessment Answer Key Form 2D Page 45 Page 46 1. y x 0 11. logistic 12. 3.138 13. 1.148 y 2. 14. x 0 15. 16. 3 log13 2 + 10 log13 x - log13 y 4 − 3 $11,778.72 3 log16 8 = − 4 6. -3 7. 4 8 shifted 2 units right, expanded vertically by a factor of 3, and 18. reflected in the x-axis 19. 11.18 years y 8. 0 x ln y = 0.0774x + 20. 3.8065 9. 10. Chapter 3 4.120 -0.791 B: A19 a2 logb − c Glencoe Precalculus Answers 17. 5. 3 y2 √ 395 3. 4. ln 3 Chapter 3 Assessment Answer Key Form 3 Page 47 Page 48 1. y x 0 15.9 11. 10 − 3 12. 2. 13. y 2 −8−6−4 0 2 4 6 8x about 7.1754 In 5 + 1.5 In x - y In 2 14. 5 y4 √ log − −4 −6 −8 15. 9 () 3 In − or about 4 16. -0.2877, 0 1696 3. 17. $18,281.88 4. 5. 18. logy 7 = x - 3 6. -2 7. 6 8. y 0 19. 3.889 10. 3.182 Chapter 3 g(x) = -3(4) x - 2 7.296 years x 20. 9. 0 y = 32.07(1.065)x In 3.12 − ≈ 0.38 and B: A20 3 In 0.49 − ≈ -0.24 3 Glencoe Precalculus Chapter 3 Assessment Answer Key Page 49, Extended-Response Test Sample Answers 1e. For x < 1, log5 x > log3 x. For x > 1, log5 x < log3 x. For x = 1, log5 x = log3 x. They intersect at (1, 0). y 1a. y = 3x y x 0 y = log5 x 1b. For x > 0, 5x > 3x. For x < 0, 5x < 3x. For x = 0, 5x = 3x. They intersect at (0, 1). x 0 y 1f. The graph of g(x) = log8 x is the graph of f(x) = log2 x compressed vertically by a y = 5x 1 . By the Change of Base factor of − 3 log2 x 1 , or − log2 x. Formula, log8 x = − x 0 log2 8 (3) 1 1c. The graph of f(x) = − x 3 x is the reflection 2. Sample answer: A colony of 9 bacteria doubles every minute. When will the population of the colony be 178? of g(x) = 3 in the y-axis. They intersect at (0, 1). 178 = 9 2x y log 178 - log 9 log 2 x = − or about 4.3 ⎞x y = ⎛⎝ 1 ⎠ 3 The colony will number 178 in about 4.3 minutes. x 0 3. log2 (x + 3) + log2 (x - 2) = 3 1d. The graph of g(x) = log3 x is the reflection of f(x) = 3x in the line y = x. The graphs do not intersect. log2 [(x + 3)(x - 2)] = 3 (x + 3)(x - 2) = 23 2 x +x-6=8 y Product Property Definition of logarithm Multiply. 2 x + x - 14 = 0 -1 ± √1 + 56 2 -1 ± √ 57 =− 2 -1 - √ 57 The solution x = − is 2 f(x) x= − 0 g(x ) x extraneous because it makes both logarithms undefined. (continued on the next page) Chapter 3 A21 Glencoe Precalculus Answers log 178 = log 9 + x log 2 Chapter 3 Assessment Answer Key Page 49, Extended-Response Test Sample Answers (continued) e2x - 3ex ≤ -2 4. Original inequality e2x - 3ex + 2 ≤ 0 x Add 2 to each side. x (e - 1)(e - 2) ≤ 0 Factor. Find the critical numbers. ex - 1 = 0 ex - 2 = 0 ex = 1 ex = 2 x x ln e = ln 1 ln e = ln 2 x ln 1 = ln 1 x ln 1 = ln 2 x=0 Set each factor equal to 0. Solve. One-to-One Property Inverse Property x = ln 2 ≈ 0.693 Create a sign chart. Test Y = 0.5 +- Test Y = -1 ++ 0 Test Y = 2 ++ ln 2 (0.693) Y Because f(x) is negative in the middle interval, the solution of e2x - 3x2 ≤ -2 is [0, ln 2]. Chapter 3 A22 Glencoe Precalculus Chapter 3 Assessment Answer Key Standardized Test Practice Page 50 1. A B C D 2. F G H J 3. 4. F A B G B C H C D J 9. A 10. B C D F G H J 11. A B C D 12. F G H J 13. A B C D 14. F G H J 15. A B C D 16. F G H J 17. A B C D 18. F G H J D Answers 5. A Page 51 6. F 7. A 8. F G B G Chapter 3 H C H J D J A23 Glencoe Precalculus Chapter 3 Assessment Answer Key Standardized Test Practice (continued) Page 52 19. (f ◦ g) (x) = 4 − x2 - 2x +1 -4 -2, -1, 1.5 20. removable 21. discontinuity y 22. x 0 24. 5000 -18 25. about 99.9 g 23. Sample answer: y = 2.497x + 177.88 26. Sample answer: 27. y = 2029 y 28a. 0 x D = (-4, ∞); b. R = (-∞, ∞) asymptote at x = -4; end behavior lim f(x) = -∞, + x→-4 c. lim f(x) = ∞ x→∞ d. increasing on (-4, ∞) Chapter 3 A24 Glencoe Precalculus