3.5 EXPONENTIAL AND LOGARITHMIC MODELS

Transcription

1 Section.5 Eponential and Logarithmic Models 55.5 EXPONENTIAL AND LOGARITHMIC MODELS What ou should learn Recognize the five most common tpes of models involving eponential and logarithmic functions. Use eponential growth and deca functions to model and solve real-life problems. Use Gaussian functions to model and solve real-life problems. Use logistic growth functions to model and solve real-life problems. Use logarithmic functions to model and solve real-life problems. Wh ou should learn it Eponential growth and deca models are often used to model the populations of countries. For instance, in Eercise on page, ou will use eponential growth and deca models to compare the populations of several countries. Introduction The five most common tpes of mathematical models involving eponential functions and logarithmic functions are as follows.. Eponential growth model:. Eponential deca model: ae b, ae b, b > 0 b > 0. Gaussian model:. Logistic growth model: 5. Logarithmic models: ae ( b) c a be r a b ln, a b log The basic shapes of the graphs of these functions are shown in Figure.. = e = e = e Eponential growth model Eponential deca model Gaussian model = + ln = + log Alan Becker/Stone/Gett Images = + e 5 Logistic growth model Natural logarithmic model Common logarithmic model FIGURE. You can often gain quite a bit of insight into a situation modeled b an eponential or logarithmic function b identifing and interpreting the function s asmptotes. Use the graphs in Figure. to identif the asmptotes of the graph of each function.

2 5 Chapter Eponential and Logarithmic Functions Eponential Growth and Deca Eample Online Advertising Estimates of the amounts (in billions of dollars) of U.S. online advertising spending from 007 through 0 are shown in the table. A scatter plot of the data is shown in Figure.. (Source: emarketer) Year Advertising spending An eponential growth model that approimates these data is given b S 0.e 0.0t, 7 t, where S is the amount of spending (in billions) and t 7 represents 007. Compare the values given b the model with the estimates shown in the table. According to this model, when will the amount of U.S. online advertising spending reach $0 billion? Dollars (in billions) Online Advertising Spending S t Year (7 007) FIGURE. Algebraic Solution The following table compares the two sets of advertising spending figures. Year Advertising spending Model To find when the amount of U.S. online advertising spending will reach $0 billion, let S 0 in the model and solve for t. 0.e 0.0t S Write original model. 0.e 0.0t 0 Substitute 0 for S. e 0.0t.87 Divide each side b 0.. ln e 0.0t ln.87 Take natural log of each side. 0.0t.58 Inverse Propert t. Divide each side b 0.0. According to the model, the amount of U.S. online advertising spending will reach $0 billion in 0. Now tr Eercise. Graphical Solution Use a graphing utilit to graph the model 0.e 0.0 and the data in the same viewing window. You can see in Figure.5 that the model appears to fit the data closel FIGURE.5 Use the zoom and trace features of the graphing utilit to find that the approimate value of for 0 is.. So, according to the model, the amount of U.S. online advertising spending will reach $0 billion in 0. TECHNOLOGY Some graphing utilities have an eponential regression feature that can be used to find eponential models that represent data. If ou have such a graphing utilit, tr using it to find an eponential model for the data given in Eample. How does our model compare with the model given in Eample?

3 Section.5 Eponential and Logarithmic Models 57 In Eample, ou were given the eponential growth model. But suppose this model were not given; how could ou find such a model? One technique for doing this is demonstrated in Eample. Eample Modeling Population Growth Population Fruit Flies 00 (5, 50) =.e 0.59t 00 (, 00) (, 00) 5 Time (in das) FIGURE. t In a research eperiment, a population of fruit flies is increasing according to the law of eponential growth. After das there are 00 flies, and after das there are 00 flies. How man flies will there be after 5 das? Solution Let be the number of flies at time t. From the given information, ou know that 00 when t and 00 when t. Substituting this information into the model ae bt produces 00 ae b and 00 ae b. To solve for b, solve for a in the first equation. 00 ae b Then substitute the result into the second equation. 00 ae b e b eb 00 eb 00 ln b ln b Solve for a in the first equation. Write second equation. 00 Substitute for a. Divide each side b 00. Take natural log of each side. Solve for b. Using b ln and the equation ou found for a, ou can determine that 00 a e 00 e ln 00.. Substitute ln for b. Simplif. Inverse Propert Simplif. So, with a. and b ln 0.59, the eponential growth model is ln.e 0.59t a 00 e b as shown in Figure.. This implies that, after 5 das, the population will be.e flies. Now tr Eercise 9. e b

4 58 Chapter Eponential and Logarithmic Functions Ratio 0 ( 0 ) 0 FIGURE.7 R t = 0 Carbon Dating R = 0 t = 5700 t = 9, ,000 Time (in ears) e t/8 t In living organic material, the ratio of the number of radioactive carbon isotopes (carbon ) to the number of nonradioactive carbon isotopes (carbon ) is about to 0. When organic material dies, its carbon content remains fied, whereas its radioactive carbon begins to deca with a half-life of about 5700 ears. To estimate the age of dead organic material, scientists use the following formula, which denotes the ratio of carbon to carbon present at an time t (in ears). R 8 e t 0 Carbon dating model The graph of R is shown in Figure.7. Note that R decreases as t increases. Eample Carbon Dating Estimate the age of a newl discovered fossil in which the ratio of carbon to carbon is R 0. Algebraic Solution In the carbon dating model, substitute the given value of R to obtain the following. 0 e t 8 R e t e t 8 0 ln e t 8 ln 0 t.0 8 t 8,9 Write original model. Let Multipl each side b 0. Take natural log of each side. Inverse Propert Multipl each side b 8. So, to the nearest thousand ears, the age of the fossil is about 9,000 ears. Now tr Eercise 5. R 0. Graphical Solution Use a graphing utilit to graph the formula for the ratio of carbon to carbon at an time t as 0 e 8. In the same viewing window, graph 0. Use the intersect feature or the zoom and trace features of the graphing utilit to estimate that 8,9 when 0, as shown in Figure FIGURE.8 = 0 e /8 = 0 5,000 So, to the nearest thousand ears, the age of the fossil is about 9,000 ears. The value of b in the eponential deca model ae bt determines the deca of radioactive isotopes. For instance, to find how much of an initial 0 grams of Ra isotope with a half-life of 599 ears is left after 500 ears, substitute this information into the model ae bt. 0 0e b 599 ln 599b Using the value of b found above and a 0, the amount left is 0e ln grams. b ln 599

5 Section.5 Eponential and Logarithmic Models 59 Gaussian Models As mentioned at the beginning of this section, Gaussian models are of the form ae b c. This tpe of model is commonl used in probabilit and statistics to represent populations that are normall distributed. The graph of a Gaussian model is called a bell-shaped curve. Tr graphing the normal distribution with a graphing utilit. Can ou see wh it is called a bell-shaped curve? For standard normal distributions, the model takes the form e. The average value of a population can be found from the bell-shaped curve b observing where the maimum -value of the function occurs. The -value corresponding to the maimum -value of the function represents the average value of the independent variable in this case,. Eample SAT Scores In 008, the Scholastic Aptitude Test (SAT) math scores for college-bound seniors roughl followed the normal distribution given b 0.00e 55,9, where is the SAT score for mathematics. Sketch the graph of this function. From the graph, estimate the average SAT score. (Source: College Board) Solution The graph of the function is shown in Figure.9. On this bell-shaped curve, the maimum value of the curve represents the average score. From the graph, ou can estimate that the average mathematics score for college-bound seniors in 008 was 55. SAT Scores % of population Distribution = Score FIGURE.9 Now tr Eercise 57..

6 0 Chapter Eponential and Logarithmic Functions Logistic Growth Models Decreasing rate of growth Increasing rate of growth Some populations initiall have rapid growth, followed b a declining rate of growth, as indicated b the graph in Figure.0. One model for describing this tpe of growth pattern is the logistic curve given b the function a be r where is the population size and is the time. An eample is a bacteria culture that is initiall allowed to grow under ideal conditions, and then under less favorable conditions that inhibit growth. A logistic growth curve is also called a sigmoidal curve. FIGURE.0 Eample 5 Spread of a Virus On a college campus of 5000 students, one student returns from vacation with a contagious and long-lasting flu virus. The spread of the virus is modeled b 5000, t 0.8t 0 999e where is the total number of students infected after t das. The college will cancel classes when 0% or more of the students are infected. a. How man students are infected after 5 das? b. After how man das will the college cancel classes? Algebraic Solution a. After 5 das, the number of students infected is e e b. Classes are canceled when the number infected is e 0.8t.5 e 0.8t e 0.8t ln e 0.8t ln t ln t.5 ln t 0. So, after about 0 das, at least 0% of the students will be infected, and the college will cancel classes. Now tr Eercise 59. Graphical Solution 5000 a. Use a graphing utilit to graph Use 999e 0.8. the value feature or the zoom and trace features of the graphing utilit to estimate that 5 when 5. So, after 5 das, about 5 students will be infected. b. Classes are canceled when the number of infected students is Use a graphing utilit to graph 5000 and e 0.8 in the same viewing window. Use the intersect feature or the zoom and trace features of the graphing utilit to find the point of intersection of the graphs. In Figure., ou can see that the point of intersection occurs near 0.. So, after about 0 das, at least 0% of the students will be infected, and the college will cancel classes. 000 = FIGURE. = e 0.8

7 Section.5 Eponential and Logarithmic Models Logarithmic Models Eample Magnitudes of Earthquakes Claro Cortes IV/Reuters /Landov On Ma, 008, an earthquake of magnitude 7.9 struck Eastern Sichuan Province, China. The total economic loss was estimated at 8 billion U.S. dollars. On the Richter scale, the magnitude R of an earthquake of intensit I is given b R log I I 0 where I 0 is the minimum intensit used for comparison. Find the intensit of each earthquake. (Intensit is a measure of the wave energ of an earthquake.) a. Nevada in 008: R.0 b. Eastern Sichuan, China in 008: R 7.9 Solution a. Because I 0 and R.0, ou have.0 log I Substitute for I 0 and.0 for R log I I 0.0,000,000. b. For R 7.9, ou have 7.9 log I Eponentiate each side. Inverse Propert Substitute for I 0 and 7.9 for R. t Year Population, P log I Eponentiate each side. I ,00,000. Inverse Propert Note that an increase of.9 units on the Richter scale (from.0 to 7.9) represents an increase in intensit b a factor of 79,00, ,000,000 In other words, the intensit of the earthquake in Eastern Sichuan was about 79 times as great as that of the earthquake in Nevada. Now tr Eercise. CLASSROOM DISCUSSION Comparing Population Models The populations P (in millions) of the United States for the census ears from 90 to 000 are shown in the table at the left. Least squares regression analsis gives the best quadratic model for these data as P.08t 9.07t 8.8, and the best eponential model for these data as P 8.77e 0.t. Which model better fits the data? Describe how ou reached our conclusion. (Source: U.S. Census Bureau)

8 Chapter Eponential and Logarithmic Functions.5 EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks.. An eponential growth model has the form and an eponential deca model has the form.. A logarithmic model has the form or.. Gaussian models are commonl used in probabilit and statistics to represent populations that are.. The graph of a Gaussian model is shaped, where the is the maimum -value of the graph. 5. A logistic growth model has the form.. A logistic curve is also called a curve. SKILLS AND APPLICATIONS In Eercises 7, match the function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (b) (c) (e) (d) (f) 8 COMPOUND INTEREST In Eercises 5, complete the table for a savings account in which interest is compounded continuousl. Initial Annual Time to Amount After Investment % Rate Double 0 Years 5. $000.5%. $ $750 7 r 8. $0,000 r 9. $500 $ $00 $9, % $0, % $ COMPOUND INTEREST In Eercises and, determine the principal P that must be invested at rate r, compounded monthl, so that $500,000 will be available for retirement in t ears.. r 5%, t 0. r %, t 5 COMPOUND INTEREST In Eercises 5 and, determine the time necessar for $000 to double if it is invested at interest rate r compounded (a) annuall, (b) monthl, (c) dail, and (d) continuousl. 5. r 0%. r.5% 7. e 8. e 9. log 0. e 5. ln. e In Eercises and, (a) solve for P and (b) solve for t.. A Pe rt. A P r n nt 7. COMPOUND INTEREST Complete the table for the time t (in ears) necessar for P dollars to triple if interest is compounded continuousl at rate r. r % % % 8% 0% % t 8. MODELING DATA Draw a scatter plot of the data in Eercise 7. Use the regression feature of a graphing utilit to find a model for the data.

9 Section.5 Eponential and Logarithmic Models 9. COMPOUND INTEREST Complete the table for the time t (in ears) necessar for P dollars to triple if interest is compounded annuall at rate r MODELING DATA Draw a scatter plot of the data in Eercise 9. Use the regression feature of a graphing utilit to find a model for the data.. COMPARING MODELS If $ is invested in an account over a 0-ear period, the amount in the account, where t represents the time in ears, is given b A t or A e 0.07t depending on whether the account pas simple interest at 7 % or continuous compound interest at 7%. Graph each function on the same set of aes. Which grows at a higher rate? (Remember that t is the greatest integer function discussed in Section..). COMPARING MODELS If $ is invested in an account over a 0-ear period, the amount in the account, where t represents the time in ears, is given b A 0.0 t or A t depending on whether the account pas simple interest at % or compound interest at 5 % compounded dail. Use a graphing utilit to graph each function in the same viewing window. Which grows at a higher rate? RADIOACTIVE DECAY In Eercises 8, complete the table for the radioactive isotope. Half-life Initial Amount After Isotope (ears) Quantit 000 Years. Ra g. C g 5. 9 Pu,00.g. Ra 599 g 7. C 575 g 8. 9 Pu,00 0. g In Eercises 9, find the eponential model ae b that fits the points shown in the graph or table (, 0) 8 r % % % 8% 0% % t (0, ) 5 8 ( 0, ) (, 5). POPULATION The populations P (in thousands) of Horr Count, South Carolina from 970 through 007 can be modeled b P e 0.08t where t represents the ear, with t 0 corresponding to 970. (Source: U.S. Census Bureau) (a) Use the model to complete the table. Year Population (b) According to the model, when will the population of Horr Count reach 00,000? (c) Do ou think the model is valid for long-term predictions of the population? Eplain.. POPULATION The table shows the populations (in millions) of five countries in 000 and the projected populations (in millions) for the ear 05. (Source: U.S. Census Bureau) Countr Bulgaria Canada China United Kingdom United States (a) Find the eponential growth or deca model ae bt or ae bt for the population of each countr b letting t 0 correspond to 000. Use the model to predict the population of each countr in 00. (b) You can see that the populations of the United States and the United Kingdom are growing at different rates. What constant in the equation ae bt is determined b these different growth rates? Discuss the relationship between the different growth rates and the magnitude of the constant. (c) You can see that the population of China is increasing while the population of Bulgaria is decreasing. What constant in the equation ae bt reflects this difference? Eplain.

10 Chapter Eponential and Logarithmic Functions 5. WEBSITE GROWTH The number of hits a new search-engine website receives each month can be modeled b 080e kt, where t represents the number of months the website has been operating. In the website s third month, there were 0,000 hits. Find the value of k, and use this value to predict the number of hits the website will receive after months.. VALUE OF A PAINTING The value V (in millions of dollars) of a famous painting can be modeled b V 0e kt, where t represents the ear, with t 0 corresponding to 000. In 008, the same painting was sold for $5 million. Find the value of k, and use this value to predict the value of the painting in POPULATION The populations P (in thousands) of Reno, Nevada from 000 through 007 can be modeled b P.8e kt, where t represents the ear, with t 0 corresponding to 000. In 005, the population of Reno was about 95,000. (Source: U.S. Census Bureau) (a) Find the value of k. Is the population increasing or decreasing? Eplain. (b) Use the model to find the populations of Reno in 00 and 05. Are the results reasonable? Eplain. (c) According to the model, during what ear will the population reach 500,000? 8. POPULATION The populations P (in thousands) of Orlando, Florida from 000 through 007 can be modeled b P 5.e kt, where t represents the ear, with t 0 corresponding to 000. In 005, the population of Orlando was about,90,000. (Source: U.S. Census Bureau) (a) Find the value of k. Is the population increasing or decreasing? Eplain. (b) Use the model to find the populations of Orlando in 00 and 05. Are the results reasonable? Eplain. (c) According to the model, during what ear will the population reach. million? 9. BACTERIA GROWTH The number of bacteria in a culture is increasing according to the law of eponential growth. After hours, there are 00 bacteria, and after 5 hours, there are 00 bacteria. How man bacteria will there be after hours? 50. BACTERIA GROWTH The number of bacteria in a culture is increasing according to the law of eponential growth. The initial population is 50 bacteria, and the population after 0 hours is double the population after hour. How man bacteria will there be after hours? 5. CARBON DATING (a) The ratio of carbon to carbon in a piece of wood discovered in a cave is R 8. Estimate the age of the piece of wood. (b) The ratio of carbon to carbon in a piece of paper buried in a tomb is R. Estimate the age of the piece of paper. 5. RADIOACTIVE DECAY Carbon dating assumes that the carbon dioide on Earth toda has the same radioactive content as it did centuries ago. If this is true, the amount of C absorbed b a tree that grew several centuries ago should be the same as the amount of C absorbed b a tree growing toda. A piece of ancient charcoal contains onl 5% as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal if the half-life of C is 575 ears? 5. DEPRECIATION A sport utilit vehicle that costs $,00 new has a book value of $,500 after ears. (a) Find the linear model V mt b. (b) Find the eponential model V ae kt. (c) Use a graphing utilit to graph the two models in the same viewing window. Which model depreciates faster in the first ears? (d) Find the book values of the vehicle after ear and after ears using each model. (e) Eplain the advantages and disadvantages of using each model to a buer and a seller. 5. DEPRECIATION A laptop computer that costs $50 new has a book value of $550 after ears. (a) Find the linear model V mt b. (b) Find the eponential model V ae kt. (c) Use a graphing utilit to graph the two models in the same viewing window. Which model depreciates faster in the first ears? (d) Find the book values of the computer after ear and after ears using each model. (e) Eplain the advantages and disadvantages of using each model to a buer and a seller. 55. SALES The sales S (in thousands of units) of a new CD burner after it has been on the market for t ears are modeled b S t 00 e kt. Fifteen thousand units of the new product were sold the first ear. (a) Complete the model b solving for k. (b) Sketch the graph of the model. (c) Use the model to estimate the number of units sold after 5 ears.

11 Section.5 Eponential and Logarithmic Models 5 5. LEARNING CURVE The management at a plastics factor has found that the maimum number of units a worker can produce in a da is 0. The learning curve for the number N of units produced per da after a new emploee has worked t das is modeled b N 0 e kt. After 0 das on the job, a new emploee produces 9 units. (a) Find the learning curve for this emploee (first, find the value of k). (b) How man das should pass before this emploee is producing 5 units per da? 57. IQ SCORES The IQ scores for a sample of a class of returning adult students at a small northeastern college roughl follow the normal distribution 0.0e 00 50, 70 5, where is the IQ score. (a) Use a graphing utilit to graph the function. (b) From the graph in part (a), estimate the average IQ score of an adult student. 58. EDUCATION The amount of time (in hours per week) a student utilizes a math-tutoring center roughl follows the normal distribution e , 7, where is the number of hours. (a) Use a graphing utilit to graph the function. (b) From the graph in part (a), estimate the average number of hours per week a student uses the tutoring center. 59. CELL SITES A cell site is a site where electronic communications equipment is placed in a cellular network for the use of mobile phones. The numbers of cell sites from 985 through 008 can be modeled b where t represents the ear, with t 5 corresponding to 985. (Source: CTIA-The Wireless Association) (a) Use the model to find the numbers of cell sites in the ears 985, 000, and 00. (b) Use a graphing utilit to graph the function. (c) Use the graph to determine the ear in which the number of cell sites will reach 5,000. (d) Confirm our answer to part (c) algebraicall. 0. POPULATION The populations P (in thousands) of Pittsburgh, Pennslvania from 000 through 007 can be modeled b P 7,0 950e 0.55t 0.08e t where t represents the ear, with t 0 corresponding to 000. (Source: U.S. Census Bureau) (a) Use the model to find the populations of Pittsburgh in the ears 000, 005, and 007. (b) Use a graphing utilit to graph the function. (c) Use the graph to determine the ear in which the population will reach. million. (d) Confirm our answer to part (c) algebraicall.. POPULATION GROWTH A conservation organization releases 00 animals of an endangered species into a game preserve. The organization believes that the preserve has a carring capacit of 000 animals and that the growth of the pack will be modeled b the logistic curve p t 000 9e 0.5t where t is measured in months (see figure). Endangered species population (a) Estimate the population after 5 months. (b) After how man months will the population be 500? (c) Use a graphing utilit to graph the function. Use the graph to determine the horizontal asmptotes, and interpret the meaning of the asmptotes in the contet of the problem.. SALES After discontinuing all advertising for a tool kit in 00, the manufacturer noted that sales began to drop according to the model S 500,000 0.e kt p Time (in months) where S represents the number of units sold and t represents 00. In 008, the compan sold 00,000 units. (a) Complete the model b solving for k. (b) Estimate sales in 0. t

12 Chapter Eponential and Logarithmic Functions GEOLOGY R log I I 0 In Eercises and, use the Richter scale for measuring the magnitudes of earthquakes.. Find the intensit I of an earthquake measuring R on the Richter scale (let I 0 ). (a) Southern Sumatra, Indonesia in 007, R 8.5 (b) Illinois in 008, R 5. (c) Costa Rica in 009, R.. Find the magnitude R of each earthquake of intensit I (let I 0 ). (a) I 99,500,000 (b) I 8,75,000 (c) I 7,000 INTENSITY OF SOUND In Eercises 5 8, use the following information for determining sound intensit. The level of sound, in decibels, with an intensit of I, is given b where I is an intensit of 0 0 watt per square meter, corresponding roughl to the faintest sound that can be heard b the human ear. In Eercises 5 and, find the level of sound. 5. (a) I 0 0 watt per m (quiet room) (b) I 0 5 watt per m (bus street corner) (c) I 0 8 watt per m (quiet radio) (d) I 0 0 watt per m (threshold of pain). (a) I 0 watt per m (rustle of leaves) (b) I 0 watt per m (jet at 0 meters) (c) I 0 watt per m (door slamming) (d) I 0 watt per m (siren at 0 meters) 7. Due to the installation of noise suppression materials, the noise level in an auditorium was reduced from 9 to 80 decibels. Find the percent decrease in the intensit level of the noise as a result of the installation of these materials. 8. Due to the installation of a muffler, the noise level of an engine was reduced from 88 to 7 decibels. Find the percent decrease in the intensit level of the noise as a result of the installation of the muffler. 0 log I/I 0, ph LEVELS In Eercises 9 7, use the acidit model given b ph log H, where acidit (ph) is a measure of the hdrogen ion concentration H (measured in moles of hdrogen per liter) of a solution. 9. Find the ph if H Find the ph if H Compute H for a solution in which ph Compute H for a solution in which ph.. 7. Apple juice has a ph of.9 and drinking water has a ph of 8.0. The hdrogen ion concentration of the apple juice is how man times the concentration of drinking water? 7. The ph of a solution is decreased b one unit. The hdrogen ion concentration is increased b what factor? 75. FORENSICS At 8:0 A.M., a coroner was called to the home of a person who had died during the night. In order to estimate the time of death, the coroner took the person s temperature twice. At 9:00 A.M. the temperature was 85.7 F, and at :00 A.M. the temperature was 8.8 F. From these two temperatures, the coroner was able to determine that the time elapsed since death and the bod temperature were related b the formula t 0 ln T where t is the time in hours elapsed since the person died and T is the temperature (in degrees Fahrenheit) of the person s bod. (This formula is derived from a general cooling principle called Newton s Law of Cooling. It uses the assumptions that the person had a normal bod temperature of 98. F at death, and that the room temperature was a constant 70 F. ) Use the formula to estimate the time of death of the person. 7. HOME MORTGAGE A $0,000 home mortgage for 0 ears at 7 % has a monthl pament of $89.0. Part of the monthl pament is paid toward the interest charge on the unpaid balance, and the remainder of the pament is used to reduce the principal. The amount that is paid toward the interest is u M M Pr r t and the amount that is paid toward the reduction of the principal is v M Pr r t. In these formulas, P is the size of the mortgage, r is the interest rate, M is the monthl pament, and t is the time (in ears). (a) Use a graphing utilit to graph each function in the same viewing window. (The viewing window should show all 0 ears of mortgage paments.) (b) In the earl ears of the mortgage, is the larger part of the monthl pament paid toward the interest or the principal? Approimate the time when the monthl pament is evenl divided between interest and principal reduction. (c) Repeat parts (a) and (b) for a repament period of 0 ears M $9.7. What can ou conclude?

13 Section.5 Eponential and Logarithmic Models HOME MORTGAGE The total interest u paid on a home mortgage of P dollars at interest rate r for t ears is rt u P r t. Consider a $0,000 home mortgage at 7 %. (a) Use a graphing utilit to graph the total interest function. (b) Approimate the length of the mortgage for which the total interest paid is the same as the size of the mortgage. Is it possible that some people are paing twice as much in interest charges as the size of the mortgage? 78. DATA ANALYSIS The table shows the time t (in seconds) required for a car to attain a speed of s miles per hour from a standing start. 8. The graph of f 5 is the graph of e g shifted to the right five units. e 8. The graph of a Gaussian model will never have an -intercept. 8. WRITING Use our school s librar, the Internet, or some other reference source to write a paper describing John Napier s work with logarithms. 8. CAPSTONE Identif each model as eponential, Gaussian, linear, logarithmic, logistic, quadratic, or none of the above. Eplain our reasoning. (a) (b) Speed, s Time, t (c) (d) Two models for these data are as follows. t s 5.87 ln s t s (e) (f) (a) Use the regression feature of a graphing utilit to find a linear model t and an eponential model t for the data. (b) Use a graphing utilit to graph the data and each model in the same viewing window. (c) Create a table comparing the data with estimates obtained from each model. (d) Use the results of part (c) to find the sum of the absolute values of the differences between the data and the estimated values given b each model. Based on the four sums, which model do ou think best fits the data? Eplain. EXPLORATION (g) (h) TRUE OR FALSE? In Eercises 79 8, determine whether the statement is true or false. Justif our answer. 79. The domain of a logistic growth function cannot be the set of real numbers. 80. A logistic growth function will alwas have an -intercept. PROJECT: SALES PER SHARE To work an etended application analzing the sales per share for Kohl s Corporation from 99 through 007, visit this tet s website at academic.cengage.com. (Data Source: Kohl s Corporation)

14 8 Chapter Eponential and Logarithmic Functions CHAPTER SUMMARY What Did You Learn? Eplanation/Eamples Review Eercises Recognize and evaluate eponential functions with base a (p. ). Graph eponential functions and use the One-to-One Propert (p. 7). The eponential function f with base a is denoted b f a where a > 0, a, and is an real number. = a (0, ) (0, ) = a 7 Section. Section. Recognize, evaluate, and graph eponential functions with base (p. 0). Use eponential functions to model and solve real-life problems (p. ). Recognize and evaluate logarithmic functions with base a (p. 7). Graph logarithmic functions (p. 9) and recognize, evaluate, and graph natural logarithmic functions (p. ). e One-to-One Propert: For a > 0 and a, a a if and onl if. The function f e is called the natural eponential function. (, e ) (, e ) Eponential functions are used in compound interest formulas (See Eample 8.) and in radioactive deca models. (See Eample 9.) For > 0, a > 0, and a, log a if and onl if a. The function f log a is called the logarithmic function with base a. The logarithmic function with base 0 is the common logarithmic function. It is denoted b log 0 or log. The graph of log a is a reflection of the graph of a about the line. = a (0, ) (, 0) = = log a (, e) (0, ) f() = e The function defined b f ln, > 0, is called the natural logarithmic function. Its graph is a reflection of the graph of f e about the line. (, e ( (0, ) f() = e ( (, e) (, 0), e (e, ) ( = g() = f () = ln Use logarithmic functions to model and solve real-life problems (p. ). A logarithmic function is used in the human memor model. (See Eample.) 59, 0

15 Chapter Summar 9 What Did You Learn? Eplanation/Eamples Review Eercises Use the change-of-base formula to rewrite and evaluate logarithmic epressions (p. 7). Let a, b, and be positive real numbers such that a and b. Then log a can be converted to a different base as follows. Base b log a log b log b a Base 0 log a log log a Base e log a ln ln a Section.5 Section. Section. Use properties of logarithms to evaluate, rewrite, epand, or condense logarithmic epressions (p. 8). Use logarithmic functions to model and solve real-life problems (p. 0). Solve simple eponential and logarithmic equations (p. ). Solve more complicated eponential equations (p. 5) and logarithmic equations (p. 7). Use eponential and logarithmic equations to model and solve real-life problems (p. 9). Recognize the five most common tpes of models involving eponential and logarithmic functions (p. 55). Use eponential growth and deca functions to model and solve real-life problems (p. 5). Let a be a positive number a, n be a real number, and u and v be positive real numbers.. Product Propert: log a uv log a u log a v ln uv ln u ln v. Quotient Propert: log a u v log a u log a v ln u v ln u ln v. Power Propert: log ln u n a u n n log a u, n ln u Logarithmic functions can be used to find an equation that relates the periods of several planets and their distances from the sun. (See Eample 7.) One-to-One Properties and Inverse Properties of eponential or logarithmic functions can be used to help solve eponential or logarithmic equations. To solve more complicated equations, rewrite the equations so that the One-to-One Properties and Inverse Properties of eponential or logarithmic functions can be used. (See Eamples 8.) Eponential and logarithmic equations can be used to find how long it will take to double an investment (see Eample 0) and to find the ear in which companies reached a given amount of sales. (See Eample.). Eponential growth model: ae b,. Eponential deca model: ae b,. Gaussian model: ae b c. Logistic growth model: 5. Logarithmic models: a be r a b ln, b > 0 b > 0 a b log An eponential growth function can be used to model a population of fruit flies (see Eample ) and an eponential deca function can be used to find the age of a fossil (see Eample ) , , Use Gaussian functions (p. 59), logistic growth functions (p. 0), and logarithmic functions (p. ) to model and solve real-life problems. A Gaussian function can be used to model SAT math scores for college-bound seniors. (See Eample.) A logistic growth function can be used to model the spread of a flu virus. (See Eample 5.) A logarithmic function can be used to find the intensit of an earthquake using its magnitude. (See Eample.)

16 70 Chapter Eponential and Logarithmic Functions REVIEW EXERCISES See for worked-out solutions to odd-numbered eercises.. In Eercises, evaluate the function at the indicated value of. Round our result to three decimal places.. f 0.,.5. f 0,. f 0.5,. f 78 5, 5.. f 7 0., f 5, n 5 Continuous A TABLE FOR AND P $5000, r %, t 0 ears P $500, r.5%, t 0 ears In Eercises 7, use the graph of f to describe the transformation that ields the graph of g f, f 5, f, f, f, g g 5 g g g. f 0., g 0... f, f, g g 8 In Eercises 5 0, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function. 5. f. f.5 7. f 5 8. f 5 9. f 0. f 8 5 In Eercises, use the One-to-One Propert to solve the equation for e 5 e 7. e 8 e In Eercises 5 8, evaluate f e at the indicated value of. Round our result to three decimal places In Eercises 9, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function. 9. h e 0. h e. f e. s t e t, t > 0 COMPOUND INTEREST In Eercises and, complete the table to determine the balance A for P dollars invested at rate r for t ears and compounded n times per ear. 5. WAITING TIMES The average time between incoming calls at a switchboard is minutes. The probabilit F of waiting less than t minutes until the net incoming call is approimated b the model F t e t. A call has just come in. Find the probabilit that the net call will be within (a) minute. (b) minutes. (c) 5 minutes.. DEPRECIATION After t ears, the value V of a car that originall cost $,970 is given b V t,970 t. (a) Use a graphing utilit to graph the function. (b) Find the value of the car ears after it was purchased. (c) According to the model, when does the car depreciate most rapidl? Is this realistic? Eplain. (d) According to the model, when will the car have no value?. In Eercises 7 0, write the eponential equation in logarithmic form. For eample, the logarithmic form of 8 is log e e 0 In Eercises, evaluate the function at the indicated value of without using a calculator.. f log, 000. g log 9,. g log,. f log, 8 In Eercises 5 8, use the One-to-One Propert to solve the equation for. 5. log 7 log. log 8 0 log ln 9 ln 8. ln ln In Eercises 9 5, find the domain, -intercept, and vertical asmptote of the logarithmic function and sketch its graph f log g log 7 5. f log 5 5. f log

17 Review Eercises 7 5. Use a calculator to evaluate f ln at (a). and (b) Round our results to three decimal places if necessar. 5. Use a calculator to evaluate f 5 ln at (a) e and (b). Round our results to three decimal places if necessar. In Eercises 55 58, find the domain, -intercept, and vertical asmptote of the logarithmic function and sketch its graph. 55. f ln 5. f ln 57. h ln 58. f ln 59. ANTLER SPREAD The antler spread a (in inches) and shoulder height h (in inches) of an adult male American elk are related b the model h log a 0 7. Approimate the shoulder height of a male American elk with an antler spread of 55 inches. 0. SNOW REMOVAL The number of miles s of roads cleared of snow is approimated b the model s 5 ln h, ln where h is the depth of the snow in inches. Use this model to find s when h 0 inches.. In Eercises, evaluate the logarithm using the change-of-base formula. Do each eercise twice, once with common logarithms and once with natural logarithms. Round the results to three decimal places.. log. log 00. log 5. log 0.8 In Eercises 5 8, use the properties of logarithms to rewrite and simplif the logarithmic epression. 5. log 8. log 7. ln 0 8. ln e In Eercises 9 7, use the properties of logarithms to epand the epression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) 9. log 70. log log ln z 7. h 5 log 7 ln, > In Eercises 75 80, condense the epression to the logarithm of a single quantit. 75. log 5 log 7. log log z 77. ln ln 78. ln ln 79. log log ln ln ln 8. CLIMB RATE The time t (in minutes) for a small plane to climb to an altitude of h feet is modeled b t 50 log 8,000 8,000 h, where 8,000 feet is the plane s absolute ceiling. (a) Determine the domain of the function in the contet of the problem. (b) Use a graphing utilit to graph the function and identif an asmptotes. (c) As the plane approaches its absolute ceiling, what can be said about the time required to increase its altitude? (d) Find the time for the plane to climb to an altitude of 000 feet. 8. HUMAN MEMORY MODEL Students in a learning theor stud were given an eam and then retested monthl for months with an equivalent eam. The data obtained in the stud are given as the ordered pairs t, s, where t is the time in months after the initial eam and s is the average score for the class. Use these data to find a logarithmic equation that relates t and s., 8.,, 78.,, 7.,, 8.5, 5, 7.,, 5.. In Eercises 8 88, solve for e 8. log 87. ln 88. ln. In Eercises 89 9, solve the eponential equation algebraicall. Approimate our result to three decimal places. 89. e e 90. e e e 8 0 In Eercises 9 and 9, use a graphing utilit to graph and solve the equation. Approimate the result to three decimal places. 9. 5e e In Eercises 95 0, solve the logarithmic equation algebraicall. Approimate the result to three decimal places. 95. ln ln ln ln 98. ln ln ln 00. ln 8

18 7 Chapter Eponential and Logarithmic Functions log 8 log 8 log 8 log log log 5 0. log 0. log In Eercises 05 08, use a graphing utilit to graph and solve the equation. Approimate the result to three decimal places. 05. ln 0 0. log log ln log e COMPOUND INTEREST You deposit $8500 in an account that pas.5% interest, compounded continuousl. How long will it take for the mone to triple? 0. METEOROLOGY The speed of the wind S (in miles per hour) near the center of a tornado and the distance d (in miles) the tornado travels are related b the model S 9 log d 5. On March 8, 95, a large tornado struck portions of Missouri, Illinois, and Indiana with a wind speed at the center of about 8 miles per hour. Approimate the distance traveled b this tornado..5 In Eercises, match the function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (b) (c) (e) e. e. ln. 7 log (d) (f) e. e In Eercises 7 and 8, find the eponential model ae b that passes through the points. 7. 0,,, 8. 0,, 5, 5 9. POPULATION In 007, the population of Florida residents aged 5 and over was about.0 million. In 05 and 00, the populations of Florida residents aged 5 and over are projected to be about. million and 5. million, respectivel. An eponential growth model that approimates these data is given b P.e 0.08t, 7 t 0, where P is the population (in millions) and t 7 represents 007. (Source: U.S. Census Bureau) (a) Use a graphing utilit to graph the model and the data in the same viewing window. Is the model a good fit for the data? Eplain. (b) According to the model, when will the population of Florida residents aged 5 and over reach 5.5 million? Does our answer seem reasonable? Eplain. 0. WILDLIFE POPULATION A species of bat is in danger of becoming etinct. Five ears ago, the total population of the species was 000. Two ears ago, the total population of the species was 00. What was the total population of the species one ear ago?. TEST SCORES The test scores for a biolog test follow a normal distribution modeled b 0.099e 7 8, 0 00, where is the test score. Use a graphing utilit to graph the equation and estimate the average test score.. TYPING SPEED In a tping class, the average number N of words per minute tped after t weeks of lessons was found to be N 57 5.e 0.t. Find the time necessar to tpe (a) 50 words per minute and (b) 75 words per minute.. SOUND INTENSITY The relationship between the number of decibels and the intensit of a sound I in watts per square meter is Find I for each decibel level. (a) (b) (c) 0 EXPLORATION 5 0 log I 0.. Consider the graph of e kt. Describe the characteristics of the graph when k is positive and when k is negative. TRUE OR FALSE? In Eercises 5 and, determine whether the equation is true or false. Justif our answer. 5. log b b. ln ln ln

19 Chapter Test 7 CHAPTER TEST See for worked-out solutions to odd-numbered eercises. Take this test as ou would take a test in class. When ou are finished, check our work against the answers given in the back of the book. In Eercises, evaluate the epression. Approimate our result to three decimal places In Eercises 5 7, construct a table of values. Then sketch the graph of the function. 5. f 0. f 7. f e 8. Evaluate (a) log and (b). ln e e 7 0 In Eercises 9, construct a table of values. Then sketch the graph of the function. Identif an asmptotes. 9. f log 0. f ln. f ln In Eercises, evaluate the logarithm using the change-of-base formula. Round our result to three decimal places.. log 7. log 0.. log In Eercises 5 7, use the properties of logarithms to epand the epression as a sum, difference, and/or constant multiple of logarithms. e. 5. log. ln 5 a 7. log z Eponential Growth In Eercises 8 0, condense the epression to the logarithm of a single quantit. 8. log log 9. ln ln 0. ln ln ln,000 (9,,77) 0,000 8,000,000,000,000 (0, 75) 8 0 FIGURE FOR 7 t In Eercises, solve the equation algebraicall. Approimate our result to three decimal places e e 5. ln 5. 8 ln 7. log log 5 7. Find an eponential growth model for the graph shown in the figure. 8. The half-life of radioactive actinium 7 Ac is.77 ears. What percent of a present amount of radioactive actinium will remain after 9 ears? 9. A model that can be used for predicting the height H (in centimeters) of a child based on his or her age is H ln,, where is the age of the child in ears. (Source: Snapshots of Applications in Mathematics) (a) Construct a table of values. Then sketch the graph of the model. (b) Use the graph from part (a) to estimate the height of a four-ear-old child. Then calculate the actual height using the model.

20 7 Chapter Eponential and Logarithmic Functions CUMULATIVE TEST FOR CHAPTERS See for worked-out solutions to odd-numbered eercises. FIGURE FOR Take this test as ou would take a test in class. When ou are finished, check our work against the answers given in the back of the book.. Plot the points, 5 and,. Find the coordinates of the midpoint of the line segment joining the points and the distance between the points. In Eercises, graph the equation without using a graphing utilit Find an equation of the line passing through and, 8.. Eplain wh the graph at the left does not represent as a function of. 7. Evaluate (if possible) the function given b f for each value. (a) f (b) f (c) f s 8. Compare the graph of each function with the graph of. (Note: It is not necessar to sketch the graphs.) (a) r (b) h (c) g In Eercises 9 and 0, find (a) f g, (b) f g, (c) fg, and (d) f/g. What is the domain of f/g? 9. f, g 0. f, g In Eercises and, find (a) f g and (b) g f. Find the domain of each composite function.. f, g. f, g 9,. Determine whether h 5 has an inverse function. If so, find the inverse function.. The power P produced b a wind turbine is proportional to the cube of the wind speed S. A wind speed of 7 miles per hour produces a power output of 750 kilowatts. Find the output for a wind speed of 0 miles per hour. 5. Find the quadratic function whose graph has a verte at 8, 5 and passes through the point, 7. In Eercises 8, sketch the graph of the function without the aid of a graphing utilit.. h 7. f t t t 8. g s s s 0 In Eercises 9, find all the zeros of the function and write the function as a product of linear factors. 9. f 8 0. f. f 0 0

21 Cumulative Test for Chapters 75. Use long division to divide b.. Use snthetic division to divide 5 b.. Use the Intermediate Value Theorem and a graphing utilit to find intervals one unit in length in which the function g is guaranteed to have a zero. Approimate the real zeros of the function. In Eercises 5 7, sketch the graph of the rational function b hand. Be sure to identif all intercepts and asmptotes. 5. f. 7. f 9 8 f Year TABLE FOR Sales, S In Eercises 8 and 9, solve the inequalit. Sketch the solution set on the real number line In Eercises 0 and, use the graph of f to describe the transformation that ields the graph of g. 0. f 5, g 5. f., g. In Eercises 5, use a calculator to evaluate the epression. Round our result to three decimal places.. log 98. log 7. ln 5. ln 0 5. Use the properties of logarithms to epand ln where >., 7. Write ln ln 5 as a logarithm of a single quantit. In Eercises 8 0, solve the equation algebraicall. Approimate the result to three decimal places. 8. e 7 9. e e 0 0. ln. The sales S (in billions of dollars) of lotter tickets in the United States from 997 through 007 are shown in the table. (Source: TLF Publications, Inc.) (a) Use a graphing utilit to create a scatter plot of the data. Let t represent the ear, with t 7 corresponding to 997. (b) Use the regression feature of the graphing utilit to find a cubic model for the data. (c) Use the graphing utilit to graph the model in the same viewing window used for the scatter plot. How well does the model fit the data? (d) Use the model to predict the sales of lotter tickets in 05. Does our answer seem reasonable? Eplain.. The number N of bacteria in a culture is given b the model N 75e kt, where t is the time in hours. If N 0 when t 8, estimate the time required for the population to double in size.

22 PROOFS IN MATHEMATICS Each of the following three properties of logarithms can be proved b using properties of eponential functions. Slide Rules The slide rule was invented b William Oughtred (57 0) in 5. The slide rule is a computational device with a sliding portion and a fied portion. A slide rule enables ou to perform multiplication b using the Product Propert of Logarithms. There are other slide rules that allow for the calculation of roots and trigonometric functions. Slide rules were used b mathematicians and engineers until the invention of the hand-held calculator in 97. Properties of Logarithms (p. 8) Let a be a positive number such that a, and let n be a real number. If u and v are positive real numbers, the following properties are true. Logarithm with Base a Natural Logarithm. Product Propert: log a uv log a u log a v ln uv ln u ln v. Quotient Propert: u a v log a u log a v ln u ln u ln v v. Power Propert: log a u n n log a u ln u n n ln u Proof Let log a u and log a v. The corresponding eponential forms of these two equations are a u and a v. To prove the Product Propert, multipl u and v to obtain uv a a a. The corresponding logarithmic form of uv a is log a uv. So, log a uv log a u log a v. To prove the Quotient Propert, divide u b v to obtain u a v a a. u u The corresponding logarithmic form of a is log a. So, v v u log a v log a u log a v. To prove the Power Propert, substitute a for u in the epression log a u n, as follows. log a u n log a a n Substitute a for u. log a a n n n log a u Propert of Eponents Inverse Propert of Logarithms Substitute log a u for. So, log a u n n log a u. 7