EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Transcription

1 4 George Marks/Retrofile/Gett Images C H A P T E R EXPONENTIAL AND LOGARITHMIC FUNCTIONS 4. Eponential Functions 4.2 The Natural Eponential Function 4.3 Logarithmic Functions 4.4 Laws of Logarithms 4.5 Eponential and Logarithmic Equations 4.6 Modeling with Eponential and Logarithmic Functions FOCUS ON MODELING Fitting Eponential and Power Curves to Data In this chapter we stud a class of functions called eponential functions. These are functions, like f 2 2, where the independent variable is in the eponent. Eponential functions are used in modeling man real-world phenomena, such as the growth of a population or the growth of an investment that earns compound interest. Once an eponential model is obtained, we can use the model to predict population size or calculate the amount of an investment for an future date. To find out when a population will reach a certain level, we use the inverse functions of eponential functions, called logarithmic functions. So if we have an eponential model for population growth, we can answer questions like: When will m cit be as crowded as the New York Cit street pictured above? 30

2 302 CHAPTER 4 Eponential and Logarithmic Functions 4. EXPONENTIAL FUNCTIONS Eponential Functions Graphs of Eponential Functions Compound Interest The Laws of Eponents are listed on page 4. In this chapter we stud a new class of functions called eponential functions. For eample, is an eponential function (with base 2). Notice how quickl the values of this function increase: Compare this with the function g2 2, where g The point is that when the variable is in the eponent, even a small change in the variable can cause a dramatic change in the value of the function. Eponential Functions f2 2 f f f ,073,74,824 To stud eponential functions, we must first define what we mean b the eponential epression a when is an real number. In Section.2 we defined a for a 0 and a rational number, but we have not et defined irrational powers. So what is meant b 5 3 or 2 p? To define a when is irrational, we approimate b rational numbers. For eample, since is an irrational number, we successivel approimate a.7, a.73, a.732, a.7320, a.73205,... b the following rational powers: Intuitivel, we can see that these rational powers of a are getting closer and closer to. It can be shown b using advanced mathematics that there is eactl one number that these powers approach. We define a 3 to be this number. For eample, using a calculator, we find The more decimal places of 3 we use in our calculation, the better our approimation of 5 3. It can be proved that the Laws of Eponents are still true when the eponents are real numbers. a 3 a 3 EXPONENTIAL FUNCTIONS The eponential function with base a is defined for all real numbers b where a 0 and a. f2 a We assume that a because the function f2 is just a constant function. Here are some eamples of eponential functions: f2 2 g2 3 h2 0 Base 2 Base 3 Base 0

3 EXAMPLE Evaluating Eponential Functions Let f2 3, and evaluate the following: (a) (c) f22 fp2 (b) (d) 2 f 32 f 22 SOLUTION We use a calculator to obtain the values of f. SECTION 4. Eponential Functions 303 Calculator kestrokes Output (a) f ^ 2 ENTER 9 (b) fa 2 3 B 3 2/ ^ ( ( _ ) 2 3 ) ENTER (c) fp2 3 p ^ P ENTER (d) fa2 B ^ 2 ENTER NOW TRY EXERCISE 5 Graphs of Eponential Functions We first graph eponential functions b plotting points. We will see that the graphs of such functions have an easil recognizable shape. EXAMPLE 2 Draw the graph of each function. (a) f2 3 Graphing Eponential Functions b Plotting Points (b) g2 a 3 b SOLUTION We calculate values of f2 and g2 and plot points to sketch the graphs in Figure. f 2 3 g2 A 3 B =3 0 FIGURE Notice that g2 a 3 b 3 3 f 2 Reflecting graphs is eplained in Section 2.5. so we could have obtained the graph of g from the graph of f b reflecting in the -ais. NOW TRY EXERCISE 5

4 304 CHAPTER 4 Eponential and Logarithmic Functions To see just how quickl f2 2 increases, let s perform the following thought eperiment. Suppose we start with a piece of paper that is a thousandth of an inch thick, and we fold it in half 50 times. Each time we fold the paper, the thickness of the paper stack doubles, so the thickness of the resulting stack would be 2 50 /000 inches. How thick do ou think that is? It works out to be more than 7 million miles! Figure 2 shows the graphs of the famil of eponential functions f2 a for various values of the base a. All of these graphs pass through the point 0, 2 because a 0 for a 0. You can see from Figure 2 that there are two kinds of eponential functions: If 0 a, the eponential function decreases rapidl. If a, the function increases rapidl (see the margin note) =0 2 =5 =3 =2 FIGURE 2 functions A famil of eponential 0 See Section 3.7, page 278, where the arrow notation used here is eplained. The -ais is a horizontal asmptote for the eponential function f2 a. This is because when a, we have a 0 as q, and when 0 a, we have a 0 as q (see Figure 2). Also, a 0 for all, so the function f2 a has domain and range 0, q 2. These observations are summarized in the following bo. GRAPHS OF EXPONENTIAL FUNCTIONS The eponential function f2 a a 0, a 2 has domain and range 0, q 2. The line 0 (the -ais) is a horizontal asmptote of f. The graph of f has one of the following shapes. (0, ) 0 Ï=a for a> (0, ) 0 Ï=a for 0<a< EXAMPLE 3 Identifing Graphs of Eponential Functions Find the eponential function f2 a whose graph is given. (a) (b) (2, 25) 5 _ 0 2 _3 0 3!3, 8@

5 SECTION 4. Eponential Functions 305 SOLUTION (a) Since f22 a 2 25, we see that the base is a 5. So f2 5. (b) Since f32 a 3, we see that the base is a. So f2 A 2B 8 2. NOW TRY EXERCISE 9 In the net eample we see how to graph certain functions, not b plotting points, but b taking the basic graphs of the eponential functions in Figure 2 and appling the shifting and reflecting transformations of Section 2.5. EXAMPLE 4 Transformations of Eponential Functions Use the graph of f2 2 to sketch the graph of each function. (a) g2 2 (b) h2 2 (c) k2 2 Shifting and reflecting of graphs is eplained in Section 2.5. SOLUTION (a) To obtain the graph of g2 2, we start with the graph of f2 2 and shift it upward unit. Notice from Figure 3(a) that the line is now a horizontal asmptote. (b) Again we start with the graph of f2 2, but here we reflect in the -ais to get the graph of h2 2 shown in Figure 3(b). (c) This time we start with the graph of f2 2 and shift it to the right b unit to get the graph of k2 2 shown in Figure 3(c). =+2 =2 2 =2 Horizontal asmptote =2 =2 0 0 _ =_2 0 FIGURE 3 (a) (b) (c) NOW TRY EXERCISES 25, 27, AND3 EXAMPLE 5 Comparing Eponential and Power Functions Compare the rates of growth of the eponential function f2 2 and the power function g2 2 b drawing the graphs of both functions in the following viewing rectangles. (a) 30, 34 b 30, 84 (b) 30, 64 b 30, 254 (c) 30, 204 b 30, 0004

6 306 CHAPTER 4 Eponential and Logarithmic Functions SOLUTION (a) Figure 4(a) shows that the graph of g2 2 catches up with, and becomes higher than, the graph of f2 2 at 2. (b) The larger viewing rectangle in Figure 4(b) shows that the graph of f2 2 overtakes that of g2 2 when 4. (c) Figure 4(c) gives a more global view and shows that when is large, f2 2 is much larger than g Ï=2 Ï=2 = = Ï=2 = FIGURE 4 (a) (b) (c) NOW TRY EXERCISE 4 Compound Interest Eponential functions occur in calculating compound interest. If an amount of mone P, called the principal, is invested at an interest rate i per time period, then after one time period the interest is Pi, and the amount A of mone is A P Pi P i2 If the interest is reinvested, then the new principal is P i2, and the amount after another time period is A P i2 i2 P i2 2. Similarl, after a third time period the amount is A P i2 3. In general, after k periods the amount is A P i2 k Notice that this is an eponential function with base i. If the annual interest rate is r and if interest is compounded n times per ear, then in each time period the interest rate is i r/n, and there are nt time periods in t ears. This leads to the following formula for the amount after t ears. COMPOUND INTEREST Compound interest is calculated b the formula At2 P a r n b nt r is often referred to as the nominal annual interest rate. where At2 amount after t ears P principal r interest rate per ear n number of times interest is compounded per ear t number of ears EXAMPLE 6 Calculating Compound Interest A sum of $000 is invested at an interest rate of 2% per ear. Find the amounts in the account after 3 ears if interest is compounded annuall, semiannuall, quarterl, monthl, and dail.

7 SECTION 4. Eponential Functions 307 SOLUTION We use the compound interest formula with P $000, r 0.2, and t 3. Compounding n Amount after 3 ears Annual 000 a b $ Semiannual a b $48.52 Quarterl a b $ Monthl a b $ Dail a b $ NOW TRY EXERCISE 5 If an investment earns compound interest, then the annual percentage ield (APY) is the simple interest rate that ields the same amount at the end of one ear. Simple interest is studied in Section.6. EXAMPLE 7 Calculating the Annual Percentage Yield Find the annual percentage ield for an investment that earns interest at a rate of 6% per ear, compounded dail. SOLUTION After one ear, a principal P will grow to the amount The formula for simple interest is Comparing, we see that r.0683, so r Thus, the annual percentage ield is 6.83%. NOW TRY EXERCISE 57 A P a b 365 P A P r2 4. EXERCISES CONCEPTS I II. The function f2 5 is an eponential function with base ; f 22, f02, f22, and f Match the eponential function with its graph (a) f2 2 (b) f2 2 (c) f2 2 III IV (d) f

8 308 CHAPTER 4 Eponential and Logarithmic Functions f 2 2 graph of and shift it 3. (a) To obtain the graph of g 2 2, we start with the (upward/downward) unit. (b) To obtain the graph of h2 2, we start with the graph of f 2 2 and shift it to the (left/right) unit. 4. In the formula At2 P r n 2 nt for compound interest the letters P, r, n, and t stand for,,, and, respectivel, and At2 stands for. So if $00 is invested at an interest rate of 6% compounded quarterl, then the amount after 2 ears is _3!2, 6@ Match the eponential function with one of the graphs labeled I or II. _3 (_3, 8) 23. f f2 5 I II 0 3 SKILLS 5 0 Use a calculator to evaluate the function at the indicated values. Round our answers to three decimals. 5. f2 4 ; f0.52, f 22, f p2, fa 3B Sketch the graph of the function b making a table of values. Use a calculator if necessar. 9. f g2 8. f2 A 3B 2. h g h2 2A 4B 5 8 Graph both functions on one set of aes. 5. f2 2 and g f2 3 and g2 A 3B Find the eponential function f2 a whose graph is given _3 f2 3 5 ; f.52, f 32, fe2, fa 4B g2 A 2 3B ; g.32, g 52, g2p2, ga 2B g2 A 3 4B 2 ; g0.72, g 7/22, g/p2, ga 2 3B f2 4 f2 A 2 3B and g2 7 and g2 A 4 3B (2, 9) 0 3 _3!_, Graph the function, not b plotting points, but b starting from the graphs in Figure 2. State the domain, range, and asmptote. 25. f f g g h2 4 A 2B 30. h f f2 A 5B g h (a) Sketch the graphs of f2 2 and g (b) How are the graphs related? 38. (a) Sketch the graphs of f2 9 /2 and g2 3. (b) Use the Laws of Eponents to eplain the relationship between these graphs. 39. Compare the functions f2 3 and g2 3 b evaluating both of them for 0,, 2, 3, 4, 5, 6, 7, 8, 9, 0, 5, and 20. Then draw the graphs of f and g on the same set of aes. f h2 f2 40. If, show that 0 a 0h f2 0 b. h h 4. (a) Compare the rates of growth of the functions f2 2 and g2 5 b drawing the graphs of both functions in the following viewing rectangles. (i) 30, 54 b 30, 204 (ii) 30, 254 b 30, (iii) 30, 504 b 30, (b) Find the solutions of the equation 2 5, rounded to one decimal place.

9 SECTION 4. Eponential Functions (a) Compare the rates of growth of the functions f2 3 and g2 4 b drawing the graphs of both functions in the following viewing rectangles: (i) 3 4, 44 b 30, 204 (ii) 30, 04 b 30, (iii) 30, 204 b 30, (b) Find the solutions of the equation 3 4, rounded to two decimal places Draw graphs of the given famil of functions for c 0.25, 0.5,, 2, 4. How are the graphs related? 43. f2 c2 44. f2 2 c Find, rounded to two decimal places, (a) the intervals on which the function is increasing or decreasing and (b) the range of the function APPLICATIONS 47. Bacteria Growth A bacteria culture contains 500 bacteria initiall and doubles ever hour. (a) Find a function that models the number of bacteria after t hours. (b) Find the number of bacteria after 24 hours. 48. Mouse Population A certain breed of mouse was introduced onto a small island with an initial population of 320 mice, and scientists estimate that the mouse population is doubling ever ear. (a) Find a function that models the number of mice after t ears. (b) Estimate the mouse population after 8 ears Compound Interest An investment of $5000 is deposited into an account in which interest is compounded monthl. Complete the table b filling in the amounts to which the investment grows at the indicated times or interest rates. 49. r 4% 50. t 5 ears Time (ears) Amount Rate per ear % 2% 3% 4% 5% 6% 5. Compound Interest If $0,000 is invested at an interest rate of 3% per ear, compounded semiannuall, find the value of the investment after the given number of ears. (a) 5 ears (b) 0 ears (c) 5 ears Amount 52. Compound Interest If $2500 is invested at an interest rate of 2.5% per ear, compounded dail, find the value of the investment after the given number of ears. (a) 2 ears (b) 3 ears (c) 6 ears 53. Compound Interest If $500 is invested at an interest rate of 3.75% per ear, compounded quarterl, find the value of the investment after the given number of ears. (a) ear (b) 2 ears (c) 0 ears 54. Compound Interest If $4000 is borrowed at a rate of 5.75% interest per ear, compounded quarterl, find the amount due at the end of the given number of ears. (a) 4 ears (b) 6 ears (c) 8 ears Present Value The present value of a sum of mone is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date. 55. Find the present value of $0,000 if interest is paid at a rate of 9% per ear, compounded semiannuall, for 3 ears. 56. Find the present value of $00,000 if interest is paid at a rate of 8% per ear, compounded monthl, for 5 ears. 57. Annual Percentage Yield Find the annual percentage ield for an investment that earns 8% per ear, compounded monthl. 58. Annual Percentage Yield Find the annual percentage ield for an investment that earns 5 2% per ear, compounded quarterl. DISCOVERY DISCUSSION WRITING 59. Growth of an Eponential Function Suppose ou are offered a job that lasts one month, and ou are to be ver well paid. Which of the following methods of pament is more profitable for ou? (a) One million dollars at the end of the month (b) Two cents on the first da of the month, 4 cents on the second da, 8 cents on the third da, and, in general, 2 n cents on the nth da 60. The Height of the Graph of an Eponential Function Your mathematics instructor asks ou to sketch a graph of the eponential function f2 2 for between 0 and 40, using a scale of 0 units to one inch. What are the dimensions of the sheet of paper ou will need to sketch this graph? DISCOVERY PROJECT Eponential Eplosion In this project we eplore an eample about collecting pennies that helps us eperience how eponential growth works. You can find the project at the book companion website:

10 30 CHAPTER 4 Eponential and Logarithmic Functions 4.2 THE NATURAL EXPONENTIAL FUNCTION The Number e The Natural Eponential Function Continuousl Compounded Interest An positive number can be used as a base for an eponential function. In this section we stud the special base e, which is convenient for applications involving calculus. Garr McMichael /Photo Researchers, Inc. The Gatewa Arch in St.Louis,Missouri, is shaped in the form of the graph of a combination of eponential functions (not a parabola,as it might first appear). Specificall,it is a catenar,which is the graph of an equation of the form ae b e b 2 (see Eercise 7).This shape was chosen because it is optimal for distributing the internal structural forces of the arch. Chains and cables suspended between two points (for eample, the stretches of cable between pairs of telephone poles) hang in the shape of a catenar. The notation e was chosen b Leonhard Euler (see page 266), probabl because it is the first letter of the word eponential. The Number e The number e is defined as the value that /n2 n approaches as n becomes large. (In calculus this idea is made more precise through the concept of a limit. See Chapter 3.) The table shows the values of the epression /n2 n for increasingl large values of n. It appears that, correct to five decimal places, e ; in fact, the approimate value to 20 decimal places is e It can be shown that e is an irrational number, so we cannot write its eact value in decimal form. The Natural Eponential Function n a n b n , , ,000, The number e is the base for the natural eponential function. Wh use such a strange base for an eponential function? It might seem at first that a base such as 0 is easier to work with. We will see, however, that in certain applications the number e is the best possible base. In this section we stud how e occurs in the description of compound interest. =3 =2 THE NATURAL EXPONENTIAL FUNCTION The natural eponential function is the eponential function =e f2 e with base e. It is often referred to as the eponential function. 0 FIGURE Graph of the natural eponential function Since 2 e 3, the graph of the natural eponential function lies between the graphs of 2 and 3, as shown in Figure. Scientific calculators have a special ke for the function f2 e. We use this ke in the net eample.

11 =e FIGURE 2 =e 0 EXAMPLE Evaluating the Eponential Function Evaluate each epression rounded to five decimal places. (a) (b) (c) SOLUTION We use the ke on a calculator to evaluate the eponential function. (a) e (b) 2e (c) e NOW TRY EXERCISE 3 EXAMPLE 2 Sketch the graph of each function. (a) e 3 f2 e SOLUTION 2e 0.53 Transformations of the Eponential Function (b) e X e4.8 g2 3e 0.5 SECTION 4.2 The Natural Eponential Function 3 (a) We start with the graph of e and reflect in the -ais to obtain the graph of e as in Figure 2. (b) We calculate several values, plot the resulting points, then connect the points with a smooth curve. The graph is shown in Figure 3. f 2 3e _3 FIGURE =3e NOW TRY EXERCISES 5 AND 7 EXAMPLE 3 An Eponential Model for the Spread of a Virus An infectious disease begins to spread in a small cit of population 0,000. After t das, the number of people who have succumbed to the virus is modeled b the function 0,000 t e 0.97t (a) How man infected people are there initiall (at time t 0)? (b) Find the number of infected people after one da, two das, and five das. (c) Graph the function, and describe its behavior. SOLUTION (a) Since 02 0,000/5 245e 0 2 0,000/250 8, we conclude that 8 people initiall have the disease. (b) Using a calculator, we evaluate 2, 22, and 52 and then round off to obtain the following values. Das Infected people

12 32 CHAPTER 4 Eponential and Logarithmic Functions 3000 (c) From the graph in Figure 4 we see that the number of infected people first rises slowl, then rises quickl between da 3 and da 8, and then levels off when about 2000 people are infected. NOW TRY EXERCISE FIGURE 4 0,000 t e 0.97t The graph in Figure 4 is called a logistic curve or a logistic growth model. Curves like it occur frequentl in the stud of population growth. (See Eercises ) Continuousl Compounded Interest In Eample 6 of Section 4. we saw that the interest paid increases as the number of compounding periods n increases. Let s see what happens as n increases indefinitel. If we let m n/r, then At2 P a r n b nt P ca r n/r rt n b d P ca m rt m b d Recall that as m becomes large, the quantit /m2 m approaches the number e. Thus, the amount approaches A Pe rt. This epression gives the amount when the interest is compounded at ever instant. CONTINUOUSLY COMPOUNDED INTEREST Continuousl compounded interest is calculated b the formula At2 Pe rt where At2 amount after t ears P principal r interest rate per ear t number of ears EXAMPLE 4 Calculating Continuousl Compounded Interest Find the amount after 3 ears if $000 is invested at an interest rate of 2% per ear, compounded continuousl. SOLUTION We use the formula for continuousl compounded interest with P $000, r 0.2, and t 3 to get Compare this amount with the amounts in Eample 6 of Section 4.. NOW TRY EXERCISE 3 A32 000e e 0.36 $ EXERCISES CONCEPTS 2. In the formula At2 Pe rt for continuousl compound interest,. The function f2 e is called the eponential the letters P, r, and t stand for,, and function. The number e is approimatel equal to., respectivel, and At2 stands for. So if $00 is invested at an interest rate of 6% compounded continuousl, then the amount after 2 ears is.

13 SECTION 4.2 The Natural Eponential Function 33 SKILLS 3 4 Use a calculator to evaluate the function at the indicated values. Round our answers to three decimals. 3. h2 e ; h32, h0.232, h2, h h2 e 2 ; h2, h 222, h 32, ha 2B 5 6 Complete the table of values, rounded to two decimal places, and sketch a graph of the function f2 3e f2 2e Graph the function, not b plotting points, but b starting from the graph of e in Figure. State the domain, range, and asmptote. 7. f2 e 8. e 9. e 0. f2 e. f2 e 2 2. e h2 e 3 4. g 2 e 2 5. The hperbolic cosine function is defined b cosh2 e e 2 (a) Sketch the graphs of the functions and 2 e 2 e on the same aes, and use graphical addition (see Section 2.6) to sketch the graph of cosh2. (b) Use the definition to show that cosh( ) cosh(). 6. The hperbolic sine function is defined b sinh2 e e 2 (a) Sketch the graph of this function using graphical addition as in Eercise 5. (b) Use the definition to show that sinh( ) sinh() 7. (a) Draw the graphs of the famil of functions for a 0.5,,.5, and 2. (b) How does a larger value of a affect the graph? 8 9 Find the local maimum and minimum values of the function and the value of at which each occurs. State each answer correct to two decimal places. 8. g g2 e e 3 APPLICATIONS f 2 a 2 e/a e /a Medical Drugs When a certain medical drug is administered to a patient, the number of milligrams remaining in the patient s bloodstream after t hours is modeled b How man milligrams of the drug remain in the patient s bloodstream after 3 hours? 2. Radioactive Deca A radioactive substance decas in such a wa that the amount of mass remaining after t das is given b the function where mt2 is measured in kilograms. (a) Find the mass at time t 0. (b) How much of the mass remains after 45 das? 22. Radioactive Deca Doctors use radioactive iodine as a tracer in diagnosing certain throid gland disorders. This tpe of iodine decas in such a wa that the mass remaining after t das is given b the function where mt2 is measured in grams. (a) Find the mass at time t 0. (b) How much of the mass remains after 20 das? 23. Sk Diving A sk diver jumps from a reasonable height above the ground. The air resistance she eperiences is proportional to her velocit, and the constant of proportionalit is 0.2. It can be shown that the downward velocit of the sk diver at time t is given b where t is measured in seconds and t2 is measured in feet per second (ft/s). (a) Find the initial velocit of the sk diver. (b) Find the velocit after 5 s and after 0 s. (c) Draw a graph of the velocit function t2. (d) The maimum velocit of a falling object with wind resistance is called its terminal velocit. From the graph in part (c) find the terminal velocit of this sk diver. (t)=80(-e _ º. t ) Dt2 50e 0.2t mt2 3e 0.05t mt2 6e 0.087t t2 80 e 0.2t Mitures and Concentrations A 50-gallon barrel is filled completel with pure water. Salt water with a concentration of 0.3 lb/gal is then pumped into the barrel, and the resulting miture overflows at the same rate. The amount of salt in the barrel at time t is given b Qt2 5 e 0.04t 2 where t is measured in minutes and Qt2 is measured in pounds. (a) How much salt is in the barrel after 5 min? (b) How much salt is in the barrel after 0 min? (c) Draw a graph of the function Qt2.

14 34 CHAPTER 4 Eponential and Logarithmic Functions (d) Use the graph in part (c) to determine the value that the amount of salt in the barrel approaches as t becomes large. Is this what ou would epect? 25. Logistic Growth Animal populations are not capable of unrestricted growth because of limited habitat and food supplies. Under such conditions the population follows a logistic growth model: d Pt2 ke ct where c, d, and k are positive constants. For a certain fish population in a small pond d 200, k, c 0.2, and t is measured in ears. The fish were introduced into the pond at time t 0. (a) How man fish were originall put in the pond? (b) Find the population after 0, 20, and 30 ears. (c) Evaluate Pt2 for large values of t. What value does the population approach as t q? Does the graph shown confirm our calculations? P Q(t)=5(-e _ º. º t ) Bird Population The population of a certain species of bird is limited b the tpe of habitat required for nesting. The population behaves according to the logistic growth model nt2 where t is measured in ears. (a) Find the initial bird population. (b) Draw a graph of the function nt2. (c) What size does the population approach as time goes on? 27. World Population The relative growth rate of world population has been decreasing steadil in recent ears. On the basis of this, some population models predict that world population will eventuall stabilize at a level that the planet can support. One such logistic model is Pt e 0.044t e 0.02t t where t 0 is the ear 2000 and population is measured in billions. (a) What world population does this model predict for the ear 2200? For 2300? (b) Sketch a graph of the function P for the ears 2000 to (c) According to this model, what size does the world population seem to approach as time goes on? 28. Tree Diameter For a certain tpe of tree the diameter D (in feet) depends on the tree s age t (in ears) according to the logistic growth model 5.4 Dt2 2.9e 0.0t Find the diameter of a 20-ear-old tree. D Compound Interest An investment of $7,000 is deposited into an account in which interest is compounded continuousl. Complete the table b filling in the amounts to which the investment grows at the indicated times or interest rates. 29. r 3% 30. t 0 ears Time (ears) Amount Rate per ear % 2% 3% 4% 5% 6% 3. Compound Interest If $2000 is invested at an interest rate of 3.5% per ear, compounded continuousl, find the value of the investment after the given number of ears. (a) 2 ears (b) 4 ears (c) 2 ears 32. Compound Interest If $3500 is invested at an interest rate of 6.25% per ear, compounded continuousl, find the value of the investment after the given number of ears. (a) 3 ears (b) 6 ears (c) 9 ears 33. Compound Interest If $600 is invested at an interest rate of 2.5% per ear, find the amount of the investment at the end of 0 ears for the following compounding methods. (a) Annuall (b) Semiannuall (c) Quarterl (d) Continuousl 34. Compound Interest If $8000 is invested in an account for which interest is compounded continuousl, find the amount of the investment at the end of 2 ears for the following interest rates. (a) 2% (b) 3% (c) 4.5% (d) 7% t Amount

15 SECTION 4.3 Logarithmic Functions Compound Interest Which of the given interest rates and compounding periods would provide the best investment? (a) 2 % per ear, compounded semiannuall (b) 2 2 4% per ear, compounded monthl (c) 2% per ear, compounded continuousl 36. Compound Interest Which of the given interest rates and compounding periods would provide the better investment? (a) 5 8% per ear, compounded semiannuall (b) 5% per ear, compounded continuousl 37. Investment A sum of $5000 is invested at an interest rate of 9% per ear, compounded continuousl. (a) Find the value At2 of the investment after t ears. (b) Draw a graph of At2. (c) Use the graph of At2 to determine when this investment will amount to $25,000. DISCOVERY DISCUSSION WRITING 38. The Definition of e Illustrate the definition of the number e b graphing the curve /2 and the line e on the same screen, using the viewing rectangle 30, 404 b 30, LOGARITHMIC FUNCTIONS Logarithmic Functions Graphs of Logarithmic Functions Common Logarithms Natural Logarithms FIGURE one-to-one. 0 f2 a is f()=a, a> In this section we stud the inverses of eponential functions. Logarithmic Functions Ever eponential function f2 a, with a 0 and a, is a one-to-one function b the Horizontal Line Test (see Figure for the case a ) and therefore has an inverse function. The inverse function f is called the logarithmic function with base a and is denoted b log a. Recall from Section 2.6 that f is defined b f 2 3 f2 This leads to the following definition of the logarithmic function. DEFINITION OF THE LOGARITHMIC FUNCTION Let a be a positive number with a. The logarithmic function with base a, denoted b log a, is defined b We read log a as log base a of is. log a 3 a So log a is the eponent to which the base a must be raised to give. B tradition the name of the logarithmic function is log a, not just a single letter. Also, we usuall omit the parentheses in the function notation and write log a 2 log a When we use the definition of logarithms to switch back and forth between the logarithmic form log a and the eponential form a, it is helpful to notice that, in both forms, the base is the same: Logarithmic form Eponent Eponential form Eponent log a Base a Base

16 36 CHAPTER 4 Eponential and Logarithmic Functions EXAMPLE Logarithmic and Eponential Forms The logarithmic and eponential forms are equivalent equations: If one is true, then so is the other. So we can switch from one form to the other as in the following illustrations. Logarithmic form Eponential form log 0 00, ,000 log log log 5 s r 5 r s NOW TRY EXERCISE 5 log Inverse Function Propert: f f22 ff 22 It is important to understand that log a is an eponent. For eample, the numbers in the right column of the table in the margin are the logarithms (base 0) of the numbers in the left column. This is the case for all bases, as the following eample illustrates. EXAMPLE 2 Evaluating Logarithms (a) log because (b) log because (c) log 0 0. because 0 0. (d) log because 6 /2 4 NOW TRY EXERCISES 7 AND 9 When we appl the Inverse Function Propert described on page 20 to f2 a and f 2 log a, we get log a a 2, a log a, 0 We list these and other properties of logarithms discussed in this section. PROPERTIES OF LOGARITHMS Propert Reason. log a 0 We must raise a to the power 0 to get. 2. log a a We must raise a to the power to get a. 3. log a a We must raise a to the power to get a. 4. a log a log a is the power to which a must be raised to get. EXAMPLE 3 Appling Properties of Logarithms We illustrate the properties of logarithms when the base is 5. log 5 0 Propert log 5 5 Propert 2 log Propert 3 5 log Propert 4 NOW TRY EXERCISES 9 AND 25

17 SECTION 4.3 Logarithmic Functions 37 = =a, a> =log a Graphs of Logarithmic Functions Recall that if a one-to-one function f has domain A and range B, then its inverse function f has domain B and range A. Since the eponential function f2 a with a has domain and range 0, q 2, we conclude that its inverse function, f 2 log a, has domain 0, q 2 and range. The graph of f 2 log is obtained b reflecting the graph of f2 a a in the line. Figure 2 shows the case a. The fact that a (for a ) is a ver rapidl increasing function for 0 implies that log a is a ver slowl increasing function for (see Eercise 92). Since log a 0, the -intercept of the function log a is. The -ais is a vertical asmptote of log a because log a q as 0. FIGURE 2 Graph of the logarithmic function f2 log a EXAMPLE 4 Sketch the graph of f2 log 2. Graphing a Logarithmic Function b Plotting Points SOLUTION To make a table of values, we choose the -values to be powers of 2 so that we can easil find their logarithms. We plot these points and connect them with a smooth curve as in Figure 3. log _3 _4 f()=log FIGURE 3 NOW TRY EXERCISE 4 Figure 4 shows the graphs of the famil of logarithmic functions with bases 2, 3, 5, and 0. These graphs are drawn b reflecting the graphs of 2, 3, 5, and 0 (see Figure 2 in Section 4.) in the line. We can also plot points as an aid to sketching these graphs, as illustrated in Eample 4. =log 2 =log =logfi =log 0 FIGURE 4 functions A famil of logarithmic

18 38 CHAPTER 4 Eponential and Logarithmic Functions MATHEMATICS IN THE MODERN WORLD Bettmann /CORBIS Hulton-Deutsch Collection /CORBIS Law Enforcement Mathematics aids law enforcement in numerous and surprising was, from the reconstruction of bullet trajectories to determining the time of death to calculating the probabilit that a DNA sample is from a particular person. One interesting use is in the search for missing persons. A person who has been missing for several ears might look quite different from his or her most recent available photograph. This is particularl true if the missing person is a child. Have ou ever wondered what ou will look like 5, 0, or 5 ears from now? Researchers have found that different parts of the bod grow at different rates. For eample, ou have no doubt noticed that a bab s head is much larger relative to its bod than an adult s. As another eample, the ratio of arm length to height is 3 in a child but 2 about 5 in an adult. B collecting data and analzing the graphs, researchers are able to determine the functions that model growth. As in all growth phenomena, eponential and logarithmic functions pla a crucial role. For instance, the formula that relates arm length l to height h is l ae kh where a and k are constants. B studing various phsical characteristics of a person, mathematical biologists model each characteristic b a function that describes how it changes over time. Models of facial characteristics can be programmed into a computer to give a picture of how a person s appearance changes over time. These pictures aid law enforcement agencies in locating missing persons. In the net two eamples we graph logarithmic functions b starting with the basic graphs in Figure 4 and using the transformations of Section 2.5. EXAMPLE 5 Sketch the graph of each function. (a) g2 log 2 (b) h2 log 2 2 Reflecting Graphs of Logarithmic Functions SOLUTION (a) We start with the graph of f2 log 2 and reflect in the -ais to get the graph of g2 log 2 in Figure 5(a). (b) We start with the graph of f2 log 2 and reflect in the -ais to get the graph of h2 log 2 2 in Figure 5(b). 0 FIGURE 5 NOW TRY EXERCISE 55 EXAMPLE 6 f()=log g()=_log (a) h()=log (_) Shifting Graphs of Logarithmic Functions Find the domain of each function, and sketch the graph. (a) g2 2 log 5 (b) h2 log 0 32 f()=log _ 0 SOLUTION (a) The graph of g is obtained from the graph of f2 log 5 (Figure 4) b shifting upward 2 units (see Figure 6). The domain of f is 0, q g()=2+logfi f()=logfi (b) 0 FIGURE 6 (b) The graph of h is obtained from the graph of f2 log 0 (Figure 4) b shifting to the right 3 units (see Figure 7). The line 3 is a vertical asmptote. Since log 0 is defined onl when 0, the domain of h2 log 0 32 is , q 2

19 SECTION 4.3 Logarithmic Functions 39 Asmptote =3 f()=log h()=log (-3) Librar of Congress FIGURE JOHN NAPIER (550 67) was a Scottish landowner for whom mathematics was a hobb. We know him toda because of his ke invention: logarithms, which he published in 64 under the title A Description of the Marvelous Rule of Logarithms. In Napier s time, logarithms were used eclusivel for simplifing complicated calculations. For eample, to multipl two large numbers, we would write them as powers of 0. The eponents are simpl the logarithms of the numbers. For instance, ,872,564 The idea is that multipling powers of 0 is eas (we simpl add their eponents). Napier produced etensive tables giving the logarithms (or eponents) of numbers. Since the advent of calculators and computers, logarithms are no longer used for this purpose. The logarithmic functions, however, have found man applications, some of which are described in this chapter. Napier wrote on man topics. One of his most colorful works is a book entitled A Plaine Discover of thewhole Revelation of Saint John, in which he predicted that the world would end in the ear 700. NOW TRY EXERCISES 53 AND 57 Common Logarithms We now stud logarithms with base 0. COMMON LOGARITHM The logarithm with base 0 is called the common logarithm and is denoted b omitting the base: From the definition of logarithms we can easil find that But how do we find log 50? We need to find the eponent such that Clearl, is too small and 2 is too large. So To get a better approimation, we can eperiment to find a power of 0 closer to 50. Fortunatel, scientific calculators are equipped with a LOG ke that directl gives values of common logarithms. EXAMPLE 7 log log 0 log 0 and log 00 2 log 50 2 Evaluating Common Logarithms Use a calculator to find appropriate values of f2 log and use the values to sketch the graph. SOLUTION We make a table of values, using a calculator to evaluate the function at those values of that are not powers of 0. We plot those points and connect them b a smooth curve as in Figure 8. log FIGURE 8 2 f()=log _ NOW TRY EXERCISE 43

20 320 CHAPTER 4 Eponential and Logarithmic Functions Scientists model human response to stimuli (such as sound, light, or pressure) using logarithmic functions. For eample, the intensit of a sound must be increased manfold before we feel that the loudness has simpl doubled. The pschologist Gustav Fechner formulated the law as S k log a I I 0 b Human response to sound and light intensit is logarithmic. We stud the decibel scale in more detail in Section 4.6. where S is the subjective intensit of the stimulus, I is the phsical intensit of the stimulus, I 0 stands for the threshold phsical intensit, and k is a constant that is different for each sensor stimulus. EXAMPLE 8 Common Logarithms and Sound The perception of the loudness B (in decibels, db) of a sound with phsical intensit I (in W/m 2 ) is given b B 0 log a I I 0 b where I 0 is the phsical intensit of a barel audible sound. Find the decibel level (loudness) of a sound whose phsical intensit I is 00 times that of I 0. SOLUTION We find the decibel level B b using the fact that I 00I 0. B 0 log a I I 0 b Definition of B The loudness of the sound is 20 db. NOW TRY EXERCISE 87 0 log a 00I 0 b I 0 0 log 00 0 # 2 20 I = 00I 0 Cancel I 0 Definition of log The notation ln is an abbreviation for the Latin name logarithmus naturalis. Natural Logarithms Of all possible bases a for logarithms, it turns out that the most convenient choice for the purposes of calculus is the number e, which we defined in Section 4.2. =e NATURAL LOGARITHM The logarithm with base e is called the natural logarithm and is denoted b ln: ln log e =ln The natural logarithmic function ln is the inverse function of the natural eponential function e. Both functions are graphed in Figure 9. B the definition of inverse functions we have = FIGURE 9 Graph of the natural logarithmic function ln 3 e If we substitute a e and write ln for log e in the properties of logarithms mentioned earlier, we obtain the following properties of natural logarithms.

21 @ SECTION 4.3 Logarithmic Functions 32 PROPERTIES OF NATURAL LOGARITHMS Propert Reason. ln 0 We must raise e to the power 0 to get. 2. ln e We must raise e to the power to get e. 3. ln e We must raise e to the power to get e. 4. e ln ln is the power to which e must be raised to get. Calculators are equipped with an logarithms. LN ke that directl gives the values of natural EXAMPLE 9 (a) ln e 8 8 Evaluating the Natural Logarithm Function Definition of natural logarithm (b) ln a e 2 b ln e 2 2 Definition of natural logarithm (c) ln Use LN ke on calculator NOW TRY EXERCISE 39 EXAMPLE 0 Finding the Domain of a Logarithmic Function Find the domain of the function f2 ln SOLUTION As with an logarithmic function, ln is defined when 0. Thus, the domain of f is , 22 NOW TRY EXERCISE 63 _3 3 3 _3 FIGURE 0 ln4 2 2 EXAMPLE Drawing the Graph of a Logarithmic Function Draw the graph of the function ln4 2 2, and use it to find the asmptotes and local maimum and minimum values. SOLUTION As in Eample 0 the domain of this function is the interval 2, 22, so we choose the viewing rectangle 3 3, 34 b 3 3, 34. The graph is shown in Figure 0, and from it we see that the lines 2 and 2 are vertical asmptotes. The function has a local maimum point to the right of and a local minimum point to the left of. B zooming in and tracing along the graph with the cursor, we find that the local maimum value is approimatel.3 and this occurs when.5. Similarl (or b noticing that the function is odd), we find that the local minimum value is about.3, and it occurs when.5. NOW TRY EXERCISE 69

22 322 CHAPTER 4 Eponential and Logarithmic Functions 4.3 EXERCISES CONCEPTS. log is the eponent to which the base 0 must be raised to get 6. Logarithmic form Eponential form. So we can complete the following table for log I 2. The function f2 log 9 is the logarithm function with base. So f92, f2, f (a) , so log (b) log , so, f82, and f Match the logarithmic function with its graph. (a) f2 log 2 (b) f2 log 2 2 III /2 log (c) f2 log 2 SKILLS (d) f 2 log Complete the table b finding the appropriate logarithmic or eponential form of the equation, as in Eample Logarithmic form log 8 8 log log 8 A 8B II IV 0 2 Eponential form 8 2/ log log 4 A6B 2 log 4 A 2B 2 4 3/ Epress the equation in eponential form. 7. (a) log (b) log (a) log 0 0. (b) log (a) log (b) 0. (a) log (b). (a) ln 5 (b) ln 5 2. (a) ln 2 2 (b) ln Epress the equation in logarithmic form. 3. (a) (b) (a) (b) 8 / (a) 8 (b) (a) 4 3/ (b) (a) e 2 (b) e 3 8. (a) e 0.5 (b) e 0.5 t 9 28 Evaluate the epression. 4 5/2 32 log 2 A 8B 3 log (a) log 3 3 (b) log 3 (c) log (a) log (b) log 4 64 (c) log (a) log 6 36 (b) log 9 8 (c) log (a) log 2 32 (b) log (c) log (a) log 3 A 27 B (b) log 0 0 (c) log (a) log 5 25 (b) log 49 7 (c) log (a) (b) 3 log 3 2 log (c) e ln5 26. (a) e ln p (b) 0 log 5 log 87 (c) (a) log (b) ln e 4 (c) 28. (a) log (b) log 4 A 4 2 2B (c) log Use the definition of the logarithmic function to find. 29. (a) log 2 5 (b) log (a) log 5 4 (b) log (a) log (b) log (a) log 4 2 (b) log (a) log 0 2 (b) log 5 2 ln/e2

23 SECTION 4.3 Logarithmic Functions (a) log (b) log (a) log 6 4 (b) 36. (a) log (b) log Use a calculator to evaluate the epression, correct to four decimal places. 37. (a) log 2 (b) log 35.2 (c) 38. (a) log 50 (b) log 2 (c) log (a) ln 5 (b) ln 25.3 (c) 40. (a) ln 27 (b) ln 7.39 (c) ln Sketch the graph of the function b plotting points. 4. f2 log f2 2 log Find the function of the form log a whose graph is given Match the logarithmic function with one of the graphs labeled I or II. 49. f2 2 ln 50. f2 ln 22 I 0 5!3, 2@ 2 0 (, 2) (5, ) log g2 log 4 g2 log 0 _! 2, _@ 5. Draw the graph of 4, then use it to draw the graph of log Draw the graph of 3, then use it to draw the graph of log 3. II 0 loga 2 3B ln 32 (3, 0) 3 =2 (9, 2) Graph the function, not b plotting points, but b starting from the graphs in Figures 4 and 9. State the domain, range, and asmptote. 53. f2 log f2 log g2 log g2 ln log log log ln 2 6. ln 62. ln Find the domain of the function. 63. f2 log f2 log g2 log g2 ln h2 ln ln h2 2 log Draw the graph of the function in a suitable viewing rectangle, and use it to find the domain, the asmptotes, and the local maimum and minimum values. 69. log ln ln 72. ln ln 74. log Find the functions f g and g f and their domains. 75. f 2 2, g f 2 3, g 2 2 f 2 log 2, g 2 2 f 2 log, g Compare the rates of growth of the functions f2 ln and g2 b drawing their graphs on a common screen using the viewing rectangle 3, 304 b 3, (a) B drawing the graphs of the functions f2 ln 2 and g2 in a suitable viewing rectangle, show that even when a logarithmic function starts out higher than a root function, it is ultimatel overtaken b the root function. (b) Find, correct to two decimal places, the solutions of the equation ln A famil of functions is given. (a) Draw graphs of the famil for c, 2, 3, and 4. (b) How are the graphs in part (a) related? 8. f2 logc2 82. f2 c log A function f2 is given. (a) Find the domain of the function f. (b) Find the inverse function of f. 83. f2 log 2 log f2 lnlnln (a) Find the inverse of the function f2. 2 (b) What is the domain of the inverse function? 2

24 324 CHAPTER 4 Eponential and Logarithmic Functions APPLICATIONS 86. Absorption of Light A spectrophotometer measures the concentration of a sample dissolved in water b shining a light through it and recording the amount of light that emerges. In other words, if we know the amount of light that is absorbed, we can calculate the concentration of the sample. For a certain substance the concentration (in moles per liter) is found b using the formula where I 0 is the intensit of the incident light and I is the intensit of light that emerges. Find the concentration of the substance if the intensit I is 70% of I 0. I 0 C 2500 ln a I I 0 b I where k is a positive constant that depends on the batter. For a certain batter, k If this batter is full discharged, how long will it take to charge to 90% of its maimum charge C 0? 9. Difficult of a Task The difficult in acquiring a target (such as using our mouse to click on an icon on our computer screen) depends on the distance to the target and the size of the target. According to Fitts s Law, the inde of difficult (ID) is given b ID log2a/w2 log 2 where W is the width of the target and A is the distance to the center of the target. Compare the difficult of clicking on an icon that is 5 mm wide to clicking on one that is 0 mm wide. In each case, assume that the mouse is 00 mm from the icon. 87. Carbon Dating The age of an ancient artifact can be determined b the amount of radioactive carbon-4 remaining in it. If D 0 is the original amount of carbon-4 and D is the amount remaining, then the artifact s age A (in ears) is given b A 8267 ln a D D 0 b Find the age of an object if the amount D of carbon-4 that remains in the object is 73% of the original amount D Bacteria Colon A certain strain of bacteria divides ever three hours. If a colon is started with 50 bacteria, then the time t (in hours) required for the colon to grow to N bacteria is given b logn/502 t 3 log 2 Find the time required for the colon to grow to a million bacteria. 89. Investment The time required to double the amount of an investment at an interest rate r compounded continuousl is given b t ln 2 r Find the time required to double an investment at 6%, 7%, and 8%. 90. Charging a Batter The rate at which a batter charges is slower the closer the batter is to its maimum charge C 0. The time (in hours) required to charge a full discharged batter to a charge C is given b t k ln a C C 0 b DISCOVERY DISCUSSION WRITING 92. The Height of the Graph of a Logarithmic Function Suppose that the graph of 2 is drawn on a coordinate plane where the unit of measurement is an inch. (a) Show that at a distance 2 ft to the right of the origin the height of the graph is about 265 mi. (b) If the graph of log 2 is drawn on the same set of aes, how far to the right of the origin do we have to go before the height of the curve reaches 2 ft? 93. The Googolple A googol is 0 00, and a googolple is 0 googol. Find logloggoogol22 and loglogloggoogolple Comparing Logarithms Which is larger, log 4 7 or log 5 24? Eplain our reasoning. 95. The Number of Digits in an Integer Compare log 000 to the number of digits in 000. Do the same for 0,000. How man digits does an number between 000 and 0,000 have? Between what two values must the common logarithm of such a number lie? Use our observations to eplain wh the number of digits in an positive integer is log. (The smbol n is the greatest integer function defined in Section 2.2.) How man digits does the number 2 00 have?

25 SECTION 4.4 Laws of Logarithms LAWS OF LOGARITHMS Laws of Logarithms Epanding and Combining Logarithmic Epressions Change of Base Formula In this section we stud properties of logarithms. These properties give logarithmic functions a wide range of applications, as we will see in Section 4.6. Laws of Logarithms Since logarithms are eponents, the Laws of Eponents give rise to the Laws of Logarithms. LAWS OF LOGARITHMS Let a be a positive number, with a. Let A, B, and C be an real numbers with A 0 and B 0. Law Description. log a AB2 log a A log a B The logarithm of a product of numbers is the sum of the logarithms of the numbers. 2. log a a A The logarithm of a quotient of numbers is the difference of the logarithms of the B b log a A log a B numbers. 3. log a A C 2 C log a A The logarithm of a power of a number is the eponent times the logarithm of the number. PROOF We make use of the propert log a a from Section 4.3. Law Let log a A u and log a B. When written in eponential form, these equations become a u A and a B Thus log a AB2 log a a u a 2 log a a u 2 u log a A log a B Law 2 Using Law, we have log a A log a ca A B b B d log a a A B b log a B so log a a A B b log a A log a B Law 3 Let log a A u. Then a u A, so log a A C 2 log a a u 2 C log a a uc 2 uc C log a A EXAMPLE Evaluate each epression. (a) log 4 2 log 4 32 (b) log 2 80 log 2 5 (c) 3 log 8 Using the Laws of Logarithms to Evaluate Epressions

26 326 CHAPTER 4 Eponential and Logarithmic Functions SOLUTION (a) log 4 2 log 4 32 log 4 2 # 322 Law log Because 64 = (b) log 2 80 log 2 5 log 2 A 80 Law 2 log Because 6 = (c) 3 log 8 log 8 /3 Law 3 5 B loga 2B Propert of negative eponents 0.30 Calculator NOW TRY EXERCISES 7, 9, AND Epanding and Combining Logarithmic Epressions The Laws of Logarithms allow us to write the logarithm of a product or a quotient as the sum or difference of logarithms. This process, called epanding a logarithmic epression, is illustrated in the net eample. EXAMPLE 2 Epanding Logarithmic Epressions Use the Laws of Logarithms to epand each epression. (a) log (b) log (c) ln a ab 3 c b SOLUTION (a) log 2 62 log 2 6 log 2 Law (b) log log 5 3 log log 5 6 log 5 Law Law 3 (c) ln a ab 3 c b lnab2 ln 3 c ln a ln b ln c /3 ln a ln b 3 ln c Law 2 Law Law 3 NOW TRY EXERCISES 9, 2, AND33 The Laws of Logarithms also allow us to reverse the process of epanding that was done in Eample 2. That is, we can write sums and differences of logarithms as a single logarithm. This process, called combining logarithmic epressions, is illustrated in the net eample. EXAMPLE 3 Combining Logarithmic Epressions Combine 3 log 2 log 2 into a single logarithm. SOLUTION 3 log 2 log 2 log 3 log 2 /2 Law 3 log 3 2 /2 2 Law NOW TRY EXERCISE 47 EXAMPLE 4 Combining Logarithmic Epressions Combine 3 ln s 2 ln t 4 lnt 2 2 into a single logarithm.

27 SECTION 4.4 Laws of Logarithms 327 SOLUTION 3 ln s 2 ln t 4 lnt 2 2 ln s 3 ln t /2 lnt lns 3 t /2 2 lnt s 3 t ln a t 2 2 b 4 Law 3 Law Law 2 NOW TRY EXERCISE 49 Warning Although the Laws of Logarithms tell us how to compute the logarithm of a product or a quotient, there is no corresponding rule for the logarithm of a sum or a difference. For instance, log a 2 log a log a In fact, we know that the right side is equal to log a 2. Also, don t improperl simplif quotients or powers of logarithms. For instance, log 6 log 2 log a 6 2 b and log log 2 Logarithmic functions are used to model a variet of situations involving human behavior. One such behavior is how quickl we forget things we have learned. For eample, if ou learn algebra at a certain performance level (sa, 90% on a test) and then don t use algebra for a while, how much will ou retain after a week, a month, or a ear? Hermann Ebbinghaus ( ) studied this phenomenon and formulated the law described in the net eample. Forgetting what we ve learned depends on how long ago we learned it. EXAMPLE 5 The Law of Forgetting If a task is learned at a performance level P 0, then after a time interval t the performance level P satisfies where c is a constant that depends on the tpe of task and t is measured in months. (a) Solve for P. (b) If our score on a histor test is 90, what score would ou epect to get on a similar test after two months? After a ear? (Assume that c 0.2.) SOLUTION (a) We first combine the right-hand side. log P log P 0 c logt 2 log P log P 0 c logt 2 log P log P 0 logt 2 c Given equation Law 3 log P log P 0 t 2 c Law 2 P P 0 t 2 c Because log is one-to-one (b) Here P 0 90, c 0.2, and t is measured in months. In two months: t 2 and 90 P In one ear: t 2 and 90 P Your epected scores after two months and one ear are 72 and 54, respectivel. NOW TRY EXERCISE 69

28 328 CHAPTER 4 Eponential and Logarithmic Functions Change of Base Formula For some purposes we find it useful to change from logarithms in one base to logarithms in another base. Suppose we are given log a and want to find log b. Let log b We write this in eponential form and take the logarithm, with base a, of each side. b Eponential form log a b 2 log a Take log a of each side log a b log a Law 3 log a Divide b log a b log a b This proves the following formula. We ma write the Change of Base Formula as log b a log a b b log a So log b is just a constant multiple of log a ; the constant is. log a b CHANGE OF BASE FORMULA log b log a log a b In particular, if we put a, then log a a, and this formula becomes log b a log a b We can now evaluate a logarithm to an base b using the Change of Base Formula to epress the logarithm in terms of common logarithms or natural logarithms and then using a calculator. EXAMPLE 6 Evaluating Logarithms with the Change of Base Formula Use the Change of Base Formula and common or natural logarithms to evaluate each logarithm, correct to five decimal places. (a) log 8 5 (b) log 9 20 SOLUTION (a) We use the Change of Base Formula with b 8 and a 0: (b) We use the Change of Base Formula with b 9 and a e: log 9 20 NOW TRY EXERCISES 55 AND 57 log 8 5 log log 0 8 ln 20 ln EXAMPLE 7 Using the Change of Base Formula to Graph a Logarithmic Function Use a graphing calculator to graph f2 log 6.

29 SECTION 4.4 Laws of Logarithms Calculators don t have a ke for log 6, so we use the Change of Base For- SOLUTION mula to write 0 36 _ FIGURE f2 log 6 ln ln 6 Since calculators do have an LN ke, we can enter this new form of the function and graph it. The graph is shown in Figure. NOW TRY EXERCISE 63 f2 log 6 ln ln EXERCISES CONCEPTS. The logarithm of a product of two numbers is the same as the of the logarithms of these numbers. So log 5 25 # The logarithm of a quotient of two numbers is the same as the of the logarithms of these numbers. So log The logarithm of a number raised to a power is the same as the power the logarithm of the number. So log #. 4. (a) We can epand log a 2 to get. z b (b) We can combine 2 log log log z to get. 5. Most calculators can find logarithms with base and base. To find logarithms with different bases, we use the Formula. To find log 7 2, we write 6. True or false? We get the same answer if we do the calculation in Eercise 5 using ln in place of log. SKILLS log 7 2 log log 7 8 Evaluate the epression. 7. log log 2 60 log log 4 log log 000. log 4 92 log log 2 9 log log 2 6 log 2 5 log log 3 00 log 3 8 log log log loglog 0 0, lnln e e Use the Laws of Logarithms to epand the epression. 9. log log log log log ln z 25. log 26. log AB log 28. log log ln ab 32. ln 2 3 3r 2 s 33. log a log a a2 z b b 4 c b log 2 a b ln a 38. ln B z b log log a 3 b log 42. log 32z B ln a log a 3 4 b b Use the Laws of Logarithms to combine the epression. 45. log log 3 2 log a a 2 z 3 b 46. log 2 2 log 7 log log 2 A log 2 B 2 log 2 C 48. log log log 3 log2 2 2 log lna b2 lna b2 2 ln c 5. ln 5 2 ln 3 ln log 5 2 log 5 3 log 5 z2 log 5 B

30 330 CHAPTER 4 Eponential and Logarithmic Functions log log 4 log log a b c log a d r log a s (b) Use part (a) to show that if k 3, then doubling the area increases the number of species eightfold Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to si decimal places. Use either natural or common logarithms. 55. log log log log log log log log Use the Change of Base Formula to show that Then use this fact to draw the graph of the function f2 log Draw graphs of the famil of functions log a for a 2, e, 5, and 0 on the same screen, using the viewing rectangle 30, 54 b 3 3, 34. How are these graphs related? 65. Use the Change of Base Formula to show that 66. Simplif: log 2 52log 5 72 log 3 ln ln 3 log e ln Show that ln ln Magnitude of Stars The magnitude M of a star is a measure of how bright a star appears to the human ee. It is defined b M 2.5 log a B B 0 b where B is the actual brightness of the star and B 0 is a constant. (a) Epand the right-hand side of the equation. (b) Use part (a) to show that the brighter a star, the less its magnitude. (c) Betelgeuse is about 00 times brighter than Albiero. Use part (a) to show that Betelgeuse is 5 magnitudes less bright than Albiero. APPLICATIONS 68. Forgetting Use the Law of Forgetting (Eample 5) to estimate a student s score on a biolog test two ears after he got a score of 80 on a test covering the same material. Assume that c 0.3 and t is measured in months. 69. Wealth Distribution Vilfredo Pareto ( ) observed that most of the wealth of a countr is owned b a few members of the population. Pareto s Principle is log P log c k log W where W is the wealth level (how much mone a person has) and P is the number of people in the population having that much mone. (a) Solve the equation for P. (b) Assume that k 2., c 8000, and W is measured in millions of dollars. Use part (a) to find the number of people who have $2 million or more. How man people have $0 million or more? 70. Biodiversit Some biologists model the number of species S in a fied area A (such as an island) b the speciesarea relationship log S log c k log A where c and k are positive constants that depend on the tpe of species and habitat. (a) Solve the equation for S. DISCOVERY DISCUSSION WRITING 72. True or False? Discuss each equation and determine whether it is true for all possible values of the variables. (Ignore values of the variables for which an term is undefined.) (a) log a log b log (b) log 2 2 log 2 log 2 (c) (d) log 2 z z log 2 (e) log P2log Q2 log P log Q log a (f) log a log b log b (g) log 2 72 log 2 7 (h) log a a a a (i) log 5 a a b 2 b log 5 a 2 log 5 b log 2 log log (j) ln a b ln A A

31 SECTION 4.5 Eponential and Logarithmic Equations Find the Error What is wrong with the following argument? log 0. 2 log 0. log0.2 2 log 0.0 log 0. log Shifting, Shrinking, and Stretching Graphs of Functions Let f2 2. Show that f22 4f2, and eplain how this shows that shrinking the graph of f horizontall has the same effect as stretching it verticall. Then use the identities e 2 e 2 e and ln22 ln 2 ln to show that for g2 e a horizontal shift is the same as a vertical stretch and for h2 ln a horizontal shrinking is the same as a vertical shift. 4.5 EXPONENTIAL AND LOGARITHMIC EQUATIONS Eponential Equations Logarithmic Equations Compound Interest In this section we solve equations that involve eponential or logarithmic functions. The techniques that we develop here will be used in the net section for solving applied problems. Eponential Equations An eponential equation is one in which the variable occurs in the eponent. For eample, The variable presents a difficult because it is in the eponent. To deal with this difficult, we take the logarithm of each side and then use the Laws of Logarithms to bring down from the eponent. 2 7 ln 2 ln 7 ln 2 ln Given equation Take ln of each side Law 3 (bring down eponent) ln 7 Solve for ln Calculator Recall that Law 3 of the Laws of Logarithms sas that log a A C C log a A. The method that we used to solve 2 7 is tpical of how we solve eponential equations in general. GUIDELINES FOR SOLVING EXPONENTIAL EQUATIONS. Isolate the eponential epression on one side of the equation. 2. Take the logarithm of each side, then use the Laws of Logarithms to bring down the eponent. 3. Solve for the variable. EXAMPLE Solving an Eponential Equation Find the solution of the equation 3 2 7, rounded to si decimal places.

32 332 CHAPTER 4 Eponential and Logarithmic Functions SOLUTION We take the common logarithm of each side and use Law 3. We could have used natural logarithms instead of common logarithms. In fact, using the same steps, we get ln ln log3 2 2 log 7 22log 3 log 7 2 log 7 log 3 log 7 log Given equation Take log of each side Law 3 (bring down eponent) Divide b log 3 Subtract 2 Calculator CHECK YOUR ANSWER Substituting into the original equation and using a calculator, we get NOW TRY EXERCISE 7 EXAMPLE 2 Solving an Eponential Equation Solve the equation 8e SOLUTION We first divide b 8 to isolate the eponential term on one side of the equation. CHECK YOUR ANSWER Substituting into the original equation and using a calculator, we get 8e NOW TRY EXERCISE 9 8e 2 20 e ln e 2 ln ln 2.5 ln Given equation Divide b 8 Take ln of each side Propert of ln Divide b 2 Calculator EXAMPLE 3 Solving an Eponential Equation Algebraicall and Graphicall Solve the equation e algebraicall and graphicall. SOLUTION : Algebraic Since the base of the eponential term is e, we use natural logarithms to solve this equation. e lne ln ln ln 4 Given equation Take ln of each side Propert of ln Subtract 3 23 ln Multipl b You should check that this answer satisfies the original equation. 2

33 SECTION 4.5 Eponential and Logarithmic Equations =4 SOLUTION 2: Graphical We graph the equations e 3 2 and 4 in the same viewing rectangle as in Figure. The solutions occur where the graphs intersect. Zooming in on the point of intersection of the two graphs, we see that 0.8. =e 3_2 NOW TRY EXERCISE 0 FIGURE 2 EXAMPLE 4 An Eponential Equation of Quadratic Tpe Solve the equation e 2 e 6 0. SOLUTION To isolate the eponential term, we factor. If we let e, we get the quadratic equation which factors as e 2 e 6 0 e 2 2 e 6 0 e 32e 22 0 e 3 0 or e 2 0 Given equation Law of Eponents Factor (a quadratic in e ) Zero-Product Propert e 3 e 2 The equation e 3 leads to ln 3. But the equation e 2 has no solution because e 0 for all. Thus, ln is the onl solution. You should check that this answer satisfies the original equation. NOW TRY EXERCISE 29 EXAMPLE 5 Solving an Eponential Equation Solve the equation 3e 2 e 0. SOLUTION First we factor the left side of the equation. CHECK YOUR ANSWER 0: 302e e 0 0 3: 3 32e e 3 9e 3 9e 3 0 3e 2 e 0 3 2e or 3 0 Thus the solutions are 0 and 3. Given equation Factor out common factors Divide b e (because e 0) Zero-Product Propert NOW TRY EXERCISE 33 Radiocarbon Dating is a method archeologists use to determine the age of ancient objects. The carbon dioide in the atmosphere alwas contains a fied fraction of radioactive carbon, carbon-4 4 C2, with a half-life of about 5730 ears. Plants absorb carbon dioide from the atmosphere, which then makes its wa to animals through the food chain.thus, all living creatures contain the same fied proportions of 4 Ctononradioactive 2 C as the atmosphere. After an organism dies, it stops assimilating 4 C, and the amount of 4 C in it begins to deca eponentiall. We can then determine the time elapsed since the death of the organism b measuring the amount of 4 C left in it. For eample, if a donke bone contains 73% as much 4 C as a living donke and it died t ears ago, then b the formula for radioactive deca (Section 4.6), t ln 22/ e We solve this eponential equation to find t 2600, so the bone is about 2600 ears old.

34 334 CHAPTER 4 Eponential and Logarithmic Functions Logarithmic Equations A logarithmic equation is one in which a logarithm of the variable occurs. For eample, log To solve for, we write the equation in eponential form Eponential form Solve for Another wa of looking at the first step is to raise the base, 2, to each side of the equation. 2 log Raise 2 to each side Propert of logarithms Solve for The method used to solve this simple problem is tpical. We summarize the steps as follows. GUIDELINES FOR SOLVING LOGARITHMIC EQUATIONS. Isolate the logarithmic term on one side of the equation; ou might first need to combine the logarithmic terms. 2. Write the equation in eponential form (or raise the base to each side of the equation). 3. Solve for the variable. EXAMPLE 6 Solving Logarithmic Equations Solve each equation for. (a) ln 8 (b) log SOLUTION (a) ln 8 e 8 Given equation Eponential form Therefore, e We can also solve this problem another wa: ln 8 e ln e 8 e 8 Given equation Raise e to each side Propert of ln (b) The first step is to rewrite the equation in eponential form. log Given equation CHECK YOUR ANSWER If 7, we get log log Eponential form (or raise 2 to each side) NOW TRY EXERCISES 37 AND 4

35 SECTION 4.5 Eponential and Logarithmic Equations 335 EXAMPLE 7 Solving a Logarithmic Equation Solve the equation 4 3 log22 6. SOLUTION We first isolate the logarithmic term. This allows us to write the equation in eponential form. 4 3 log log22 2 log Given equation Subtract 4 Divide b 3 Eponential form (or raise 0 to each side) Divide b 2 CHECK YOUR ANSWER If 5000, we get 4 3 log log 0, NOW TRY EXERCISE 43 CHECK YOUR ANSWER 4: log 4 22 log 4 2 log 22 log 52 undefined 3: log3 22 log3 2 log 5 log 2 log5 # 22 log _3 FIGURE 2 EXAMPLE 8 Solving a Logarithmic Equation Algebraicall and Graphicall Solve the equation log 22 log 2 algebraicall and graphicall. SOLUTION : Algebraic We first combine the logarithmic terms, using the Laws of Logarithms. Law Eponential form (or raise 0 to each side) Epand left side Subtract 0 Factor We check these potential solutions in the original equation and find that 4 is not a solution (because logarithms of negative numbers are undefined), but 3 is a solution. (See Check Your Answers.) SOLUTION 2: Graphical We first move all terms to one side of the equation: Then we graph log or 3 as in Figure 2. The solutions are the -intercepts of the graph. Thus, the onl solution is 3. NOW TRY EXERCISE 49 log 22 log 2 0 log 22 log 2

36 336 CHAPTER 4 Eponential and Logarithmic Functions In Eample 9 it s not possible to isolate algebraicall, so we must solve the equation graphicall. EXAMPLE 9 Solving a Logarithmic Equation Graphicall Solve the equation 2 2 ln 22. SOLUTION We first move all terms to one side of the equation 2 Then we graph 2 2 ln 22 0 _ ln 22 as in Figure 3. The solutions are the -intercepts of the graph. Zooming in on the -intercepts, we see that there are two solutions: FIGURE 3 _2 NOW TRY EXERCISE and.60 Logarithmic equations are used in determining the amount of light that reaches various depths in a lake. (This information helps biologists to determine the tpes of life a lake can support.) As light passes through water (or other transparent materials such as glass or plastic), some of the light is absorbed. It s eas to see that the murkier the water, the more light is absorbed. The eact relationship between light absorption and the distance light travels in a material is described in the net eample. EXAMPLE 0 Transparenc of a Lake If I 0 and I denote the intensit of light before and after going through a material and is the distance (in feet) the light travels in the material, then according to the Beer-Lambert Law, k ln a I b I 0 The intensit of light in a lake diminishes with depth. where k is a constant depending on the tpe of material. (a) Solve the equation for I. (b) For a certain lake k 0.025, and the light intensit is I 0 4 lumens (lm). Find the light intensit at a depth of 20 ft. SOLUTION (a) We first isolate the logarithmic term. k ln a I b I 0 Given equation ln a I I 0 b k I I 0 e k I I 0 e k Multipl b k Eponential form Multipl b I 0 (b) We find I using the formula from part (a). I I 0 e k 4e From part (a) I 0 = 4, k = 0.025, = 20 Calculator The light intensit at a depth of 20 ft is about 8.5 lm. NOW TRY EXERCISE 85

37 Compound Interest Recall the formulas for interest that we found in Section 4.. If a principal P is invested at an interest rate r for a period of t ears, then the amount A of the investment is given b A P r2 SECTION 4.5 Eponential and Logarithmic Equations 337 Simple interest (for one ear) At2 P a r n b nt At2 Pe rt Interest compounded n times per ear Interest compounded continuousl We can use logarithms to determine the time it takes for the principal to increase to a given amount. EXAMPLE Finding the Term for an Investment to Double A sum of $5000 is invested at an interest rate of 5% per ear. Find the time required for the mone to double if the interest is compounded according to the following method. (a) Semiannuall (b) Continuousl SOLUTION (a) We use the formula for compound interest with P $5000, At2 $0,000, r 0.05, and n 2 and solve the resulting eponential equation for t. P a r nt 5000 a t n b A 2 b 0, t 2 Divide b 5000 log.025 2t log 2 Take log of each side 2t log.025 log 2 Law 3 (bring down the eponent) t log 2 Divide b 2 log log.025 t 4.04 Calculator The mone will double in 4.04 ears. (b) We use the formula for continuousl compounded interest with P $5000, At2 $0,000, and r 0.05 and solve the resulting eponential equation for t. 5000e 0.05t 0,000 e 0.05t 2 ln e 0.05t ln t ln 2 t ln t 3.86 The mone will double in 3.86 ears. NOW TRY EXERCISE 75 Pe rt = A Divide b 5000 Take ln of each side Propert of ln Divide b 0.05 Calculator EXAMPLE 2 Time Required to Grow an Investment A sum of $000 is invested at an interest rate of 4% per ear. Find the time required for the amount to grow to $4000 if interest is compounded continuousl.

38 338 CHAPTER 4 Eponential and Logarithmic Functions SOLUTION We use the formula for continuousl compounded interest with P $000, At2 $4000, and r 0.04 and solve the resulting eponential equation for t. 000e 0.04t 4000 Pe rt = A e 0.04t 4 Divide b t ln 4 Take ln of each side t ln 4 Divide b t Calculator The amount will be $4000 in about 34 ears and 8 months. NOW TRY EXERCISE EXERCISES CONCEPTS. Let s solve the eponential equation 2e 50. (a) First, we isolate e to get the equivalent equation. (b) Net, we take ln of each side to get the equivalent equation. (c) Now we use a calculator to find. 2. Let s solve the logarithmic equation log 3 log 22 log. (a) First, we combine the logarithms to get the equivalent equation. (b) Net, we write each side in eponential form to get the equivalent equation. (c) Now we find. SKILLS 3 28 Find the solution of the eponential equation, rounded to four decimal places e e e e 2 7. e / / e e A4B / e Solve the equation. 0 2 e t t e 2 3e e 2 e e 4 4e e 2e e e e e e Solve the logarithmic equation for. 37. ln ln log log log log log log log 2 3 log 2 log 2 5 log log log 2 log log log 2 log log 5 log 5 2 log log 5 2 log log 3 52 log log 2 log log log log 9 52 log ln 2 ln For what value of is the following true? log 32 log log For what value of is it true that log log? 57. Solve for : 2 2/log Solve for : log 2 log 3 2 4

39 SECTION 4.5 Eponential and Logarithmic Equations Use a graphing device to find all solutions of the equation, rounded to two decimal places. 59. ln log log ln e e Solve the inequalit. 67. log 22 log log e 2e Transparenc of a Lake Environmental scientists measure the intensit of light at various depths in a lake to find the transparenc of the water. Certain levels of transparenc are required for the biodiversit of the submerged macrophte population. In a certain lake the intensit of light at depth is given b I 0e where I is measured in lumens and in feet. (a) Find the intensit I at a depth of 30 ft. (b) At what depth has the light intensit dropped to I 5? 7-74 Find the inverse function of f. 7. f f f 2 log f 2 log 3 APPLICATIONS 75. Compound Interest A man invests $5000 in an account that pas 8.5% interest per ear, compounded quarterl. (a) Find the amount after 3 ears. (b) How long will it take for the investment to double? 76. Compound Interest A woman invests $6500 in an account that pas 6% interest per ear, compounded continuousl. (a) What is the amount after 2 ears? (b) How long will it take for the amount to be $8000? 77. Compound Interest Find the time required for an investment of $5000 to grow to $8000 at an interest rate of 7.5% per ear, compounded quarterl. 78. Compound Interest Nanc wants to invest $4000 in saving certificates that bear an interest rate of 9.75% per ear, compounded semiannuall. How long a time period should she choose to save an amount of $5000? 79. Doubling an Investment How long will it take for an investment of $000 to double in value if the interest rate is 8.5% per ear, compounded continuousl? 80. Interest Rate A sum of $000 was invested for 4 ears, and the interest was compounded semiannuall. If this sum amounted to $ in the given time, what was the interest rate? 8. Radioactive Deca A 5-g sample of radioactive iodine decas in such a wa that the mass remaining after t das is given b mt2 5e 0.087t, where mt2 is measured in grams. After how man das is there onl 5 g remaining? 82. Sk Diving The velocit of a sk diver t seconds after jumping is given b t2 80 e 0.2t 2. After how man seconds is the velocit 70 ft/s? 83. Fish Population A small lake is stocked with a certain species of fish. The fish population is modeled b the function P 0 4e 0.8t where P is the number of fish in thousands and t is measured in ears since the lake was stocked. (a) Find the fish population after 3 ears. (b) After how man ears will the fish population reach 5000 fish? 85. Atmospheric Pressure Atmospheric pressure P (in kilopascals, kpa) at altitude h (in kilometers, km) is governed b the formula ln a P h b P 0 k where k 7 and P 0 00 kpa are constants. (a) Solve the equation for P. (b) Use part (a) to find the pressure P at an altitude of 4 km. 86. Cooling an Engine Suppose ou re driving our car on a cold winter da (20 F outside) and the engine overheats (at about 220 F). When ou park, the engine begins to cool down. The temperature T of the engine t minutes after ou park satisfies the equation T 20 ln a 200 b 0.t (a) Solve the equation for T. (b) Use part (a) to find the temperature of the engine after 20 min t Electric Circuits An electric circuit contains a batter that produces a voltage of 60 volts (V), a resistor with a resistance of 3 ohms ( ), and an inductor with an inductance of 5 henrs (H), as shown in the figure. Using calculus, it can be shown that the current I It2 (in amperes, A) t seconds after the switch is closed is I 60 3 e 3t/5 2. (a) Use this equation to epress the time t as a function of the current I. (b) After how man seconds is the current 2 A? 60 V 3 Switch 5 H

40 340 CHAPTER 4 Eponential and Logarithmic Functions 88. Learning Curve A learning curve is a graph of a function Pt2 that measures the performance of someone learning a skill as a function of the training time t. At first, the rate of learning is rapid. Then, as performance increases and approaches a maimal value M, the rate of learning decreases. It has been found that the function Pt2 M Ce kt where k and C are positive constants and C M is a reasonable model for learning. (a) Epress the learning time t as a function of the performance level P. (b) For a pole-vaulter in training, the learning curve is given b Pt2 20 4e 0.024t where Pt2 is the height he is able to pole-vault after t months. After how man months of training is he able to vault 2 ft? (c) Draw a graph of the learning curve in part (b). DISCOVERY DISCUSSION WRITING 89. Estimating a Solution Without actuall solving the equation, find two whole numbers between which the solution of 9 20 must lie. Do the same for Eplain how ou reached our conclusions. 90. A Surprising Equation Take logarithms to show that the equation /log 5 has no solution. For what values of k does the equation /log k have a solution? What does this tell us about the graph of the function f2 /log? Confirm our answer using a graphing device. 9. Disguised Equations Each of these equations can be transformed into an equation of linear or quadratic tpe b appling the hint. Solve each equation. (a) 2 log [Take log of each side.] (b) log 2 log 4 log 8 [Change all logs to base 2.] (c) [Write as a quadratic in 2.] 4.6 MODELING WITH EXPONENTIAL AND LOGARITHMIC FUNCTIONS Eponential Growth (Doubling Time) Eponential Growth (Relative Growth Rate) Radioactive Deca Newton s Law of Cooling Logarithmic Scales Man processes that occur in nature, such as population growth, radioactive deca, heat diffusion, and numerous others, can be modeled b using eponential functions. Logarithmic functions are used in models for the loudness of sounds, the intensit of earthquakes, and man other phenomena. In this section we stud eponential and logarithmic models. Eponential Growth (Doubling Time) Suppose we start with a single bacterium, which divides ever hour. After one hour we have 2 bacteria, after two hours we have 2 2 or 4 bacteria, after three hours we have 2 3 or 8 bacteria, and so on (see Figure ). We see that we can model the bacteria population after t hours b f t2 2 t. FIGURE Bacteria population

41 SECTION 4.6 Modeling with Eponential and Logarithmic Functions 34 If we start with 0 of these bacteria, then the population is modeled b f t2 0 # 2 t. A slower-growing strain of bacteria doubles ever 3 hours; in this case the population is modeled b f t2 0 # 2 t/3. In general, we have the following. EXPONENTIAL GROWTH (DOUBLING TIME) If the intial size of a population is the population at time t is n 0 and the doubling time is a, then the size of nt2 n 0 2 t/a where a and t are measured in the same time units (minutes, hours, das, ears, and so on). EXAMPLE Bacteria Population Under ideal conditions a certain bacteria population doubles ever three hours. Initiall there are 000 bacteria in a colon. (a) Find a model for the bacteria population after t hours. (b) How man bacteria are in the colon after 5 hours? (c) When will the bacteria count reach 00,000? SOLUTION (a) The population at time t is modeled b where t is measured in hours. (b) After 5 hours the number of bacteria is (c) We set nt2 00,000 in the model that we found in part (a) and solve the resulting eponential equation for t. 00, # 2 t/3 nt2 000 # 2t/ t/3 Divide b 000 log 00 log 2 t/3 nt2 000 # 2 t/3 n # 2 5/3 32,000 Take log of each side 2 t 3 log 2 Properties of log For guidelines on working with significant figures, see the Appendi, Calculations and Significant Figures. The bacteria level reaches 00,000 in about 20 hours. NOW TRY EXERCISE t log 2 Solve for t EXAMPLE 2 Rabbit Population A certain breed of rabbit was introduced onto a small island 8 months ago. The current rabbit population on the island is estimated to be 400 and doubling ever 3 months. (a) What was the initial size of the rabbit population? (b) Estimate the population one ear after the rabbits were introduced to the island. (c) Sketch a graph of the rabbit population.

42 342 CHAPTER 4 Eponential and Logarithmic Functions SOLUTION (a) The doubling time is a 3, so the population at time t is 20, FIGURE 2 nt2 645 # 2 t/3 For guidelines on working with significant figures, see the Appendi, Calculations and Significant Figures. where n is the initial population. Since the population is 400 when t is 8 months, t/3 nt2 n 0 2 Model 0 we have From model Because n n Divide b 2 8/ and switch sides 2 8/3 n Calculate Thus we estimate that 645 rabbits were introduced onto the island. (b) From part (a) we know that the initial population is n 0 645, so we can model the population after t months b After one ear t 2, so Model So after one ear there would be about 0,000 rabbits. (c) We first note that the domain is t 0. The graph is shown in Figure 2. NOW TRY EXERCISE 3 n82 n 0 2 8/3 400 n 0 2 8/3 nt2 645 # 2 t/3 n # 2 2/3 0,320 Eponential Growth (Relative Growth Rate) We have used an eponential function with base 2 to model population growth (in terms of the doubling time). We could also model the same population with an eponential function with base 3 (in terms of the tripling time). In fact, we can find an eponential model with an base. If we use the base e, we get the following model of a population in terms of the relative growth rate r: the rate of population growth epressed as a proportion of the population at an time. For instance, if r 0.02, then at an time t the growth rate is 2% of the population at time t. EXPONENTIAL GROWTH (RELATIVE GROWTH RATE) A population that eperiences eponential growth increases according to the model where nt2 population at time t n 0 initial size of the population r relative rate of growth (epressed as a proportion of the population) t time nt2 n 0 e rt Notice that the formula for population growth is the same as that for continuousl compounded interest. In fact, the same principle is at work in both cases: The growth of a population (or an investment) per time period is proportional to the size of the population (or

43 SECTION 4.6 Modeling with Eponential and Logarithmic Functions 343 the amount of the investment). A population of,000,000 will increase more in one ear than a population of 000; in eactl the same wa, an investment of $,000,000 will increase more in one ear than an investment of $000. In the following eamples we assume that the populations grow eponentiall. EXAMPLE 3 Predicting the Size of a Population The initial bacterium count in a culture is 500. A biologist later makes a sample count of bacteria in the culture and finds that the relative rate of growth is 40% per hour. (a) Find a function that models the number of bacteria after t hours. (b) What is the estimated count after 0 hours? (c) When will the bacteria count reach 80,000? (d) Sketch the graph of the function nt n(t)=500eº FIGURE 3 6 SOLUTION (a) We use the eponential growth model with n and r 0.4 to get nt2 500e 0.4t where t is measured in hours. (b) Using the function in part (a), we find that the bacterium count after 0 hours is n02 500e e 4 27,300 (c) We set nt2 80,000 and solve the resulting eponential equation for t: nt2 500 # e 0.4t 80, # e 0.4t 60 e 0.4t Divide b 500 ln t Take ln of each side ln 60 t 2.68 Solve for t 0.4 The bacteria level reaches 80,000 in about 2.7 hours. (d) The graph is shown in Figure 3. NOW TRY EXERCISE 5 The relative growth of world population has been declining over the past few decades from 2% in 995 to.3% in Standing Room Onl The population of the world was about 6. billion in 2000 and was increasing at.4% per ear. Assuming that each person occupies an average of 4 ft 2 of the surface of the earth, the eponential model for population growth projects that b the ear 280 there will be standing room onl! (The total land surface area of the world is about ft 2.) EXAMPLE 4 Comparing Different Rates of Population Growth In 2000 the population of the world was 6. billion, and the relative rate of growth was.4% per ear. It is claimed that a rate of.0% per ear would make a significant difference in the total population in just a few decades. Test this claim b estimating the population of the world in the ear 2050 using a relative rate of growth of (a).4% per ear and (b).0% per ear. Graph the population functions for the net 00 ears for the two relative growth rates in the same viewing rectangle. SOLUTION (a) B the eponential growth model we have nt2 6.e 0.04t where nt2 is measured in billions and t is measured in ears since Because the ear 2050 is 50 ears after 2000, we find n502 6.e e The estimated population in the ear 2050 is about 2.3 billion.

44 344 CHAPTER 4 Eponential and Logarithmic Functions 30 n(t)=6.e 0.04t n(t)=6.e 0.0t 0 00 FIGURE 4 (b) We use the function and find nt2 6.e 0.00t n502 6.e e The estimated population in the ear 2050 is about 0. billion. The graphs in Figure 4 show that a small change in the relative rate of growth will, over time, make a large difference in population size. NOW TRY EXERCISE 7 EXAMPLE 5 Epressing a Model in Terms of e A culture starts with 0,000 bacteria, and the number doubles ever 40 minutes. (a) Find a function nt2 n 0 2 t/a that models the number of bacteria after t minutes. (b) Find a function nt2 n 0 e rt that models the number of bacteria after t minutes. (c) Sketch a graph of the number of bacteria at time t. SOLUTION (a) The initial population is n 0 0,000. The doubling time is a 40 min 2/3 h. Since /a 3/2.5, the model is nt2 0,000 # 2.5t (b) The initial population is n 0 0,000. We need to find the relative growth rate r. Since there are 20,000 bacteria when t 2/3 h, we have 500,000 20,000 0,000e r2/32 2 e r2/32 ln 2 ln e r2/32 ln 2 r2/32 r 3 ln nt2 0,000e rt Divide b 0,000 Take ln of each side Propert of ln Solve for r 0 4 FIGURE 5 Graphs of 0,000 # 2.5t and 0,000e.0397t Now that we know the relative growth rate r, we can find the model: nt2 0,000e.0397t (c) We can graph the model in part (a) or the one in part (b). The graphs are identical. See Figure 5. NOW TRY EXERCISE 9 Radioactive Deca Radioactive substances deca b spontaneousl emitting radiation. The rate of deca is proportional to the mass of the substance. This is analogous to population growth ecept that the mass decreases. Phsicists epress the rate of deca in terms of half-life. For eample, the half-life of radium-226 is 600 ears, so a 00-g sample decas to 50 g or g2 in 600 ears, then to 25 g or g2 in 3200 ears, and so on. In

45 SECTION 4.6 Modeling with Eponential and Logarithmic Functions 345 The half-lives of radioactive elements var from ver long to ver short. Here are some eamples. Element Half-life general, for a radioactive substance with mass at time t is modeled b m 0 mt2 m 0 2 t/h and half-life h, the amount remaining Thorium-232 Uranium-235 Thorium-230 Plutonium-239 Carbon-4 Radium-226 Cesium-37 Strontium-90 Polonium-20 Thorium-234 Iodine-35 Radon-222 Lead-2 Krpton billion ears 4.5 billion ears 80,000 ears 24,360 ears 5,730 ears,600 ears 30 ears 28 ears 40 das 25 das 8 das 3.8 das 3.6 minutes 0 seconds where h and t are measured in the same time units (minutes, hours, das, ears, and so on). To epress this model in the form mt2 m 0 e rt, we need to find the relative deca rate r. Since h is the half-life, we have mt2 m 0 e rt ln m 0 2 m 0 e rh e rh 2 2 rh r ln 2 h Model h is the half-life Divide b m 0 Take ln of each side Solve for r This last equation allows us to find the rate r from the half-life h. RADIOACTIVE DECAY MODEL If m 0 is the initial mass of a radioactive substance with half-life h, then the mass remaining at time t is modeled b the function mt2 m 0 e rt where r ln 2. h In parts (c) and (d) we can also use the model found in part (a). Check that the result is the same using either model. EXAMPLE 6 Radioactive Deca Polonium Po2 has a half-life of 40 das. Suppose a sample of this substance has a mass of 300 mg. (a) Find a function mt2 m 0 2 t/h that models the mass remaining after t das. (b) Find a function mt2 m 0 e rt that models the mass remaining after t das. (c) Find the mass remaining after one ear. (d) How long will it take for the sample to deca to a mass of 200 mg? (e) Draw a graph of the sample mass as a function of time. SOLUTION (a) We have m and h 40, so the amount remaining after t das is mt2 300 # 2 t/40 (b) We have m and r ln 2/ , so the amount remaining after t das is mt2 300 # e t (c) We use the function we found in part (a) with t 365 (one ear). m e Thus, approimatel 49 mg of 20 Po remains after one ear.

46 346 CHAPTER 4 Eponential and Logarithmic Functions (d) We use the function that we found in part (a) with mt2 200 and solve the resulting eponential equation for t. 300e t 200 e t 2 3 ln e t ln 2 3 m(t) = m 0 e rt Divided b 300 Take ln of each side Joel W. Rogers/CORBIS Radioactive Waste Harmful radioactive isotopes are produced whenever a nuclear reaction occurs, whether as the result of an atomic bomb test, a nuclear accident such as the one at Chernobl in 986, or the uneventful production of electricit at a nuclear power plant. One radioactive material that is produced in atomic bombs is the isotope strontium Sr2, with a half-life of 28 ears. This is deposited like calcium in human bone tissue, where it can cause leukemia and other cancers. However, in the decades since atmospheric testing of nuclear weapons was halted, 90 Sr levels in the environment have fallen to a level that no longer poses a threat to health. Nuclear power plants produce radioactive plutonium Pu2, which has a half-life of 24,360 ears. Because of its long half-life, 239 Pu could pose a threat to the environment for thousands of ears. So great care must be taken to dispose of it properl. The difficult of ensuring the safet of the disposed radioactive waste is one reason that nuclear power plants remain controversial. Propert of ln t Solve for t t 8.9 Calculator The time required for the sample to deca to 200 mg is about 82 das. (e) We can graph the model in part (a) or the one in part (b). The graphs are identical. See Figure 6. FIGURE 6 NOW TRY EXERCISE t ln 2 3 Amount of ºPo (mg) m(t) Newton s Law of Cooling 0 50 ln 2 3 m(t)=300 e _ t 50 Time (das) Newton s Law of Cooling states that the rate at which an object cools is proportional to the temperature difference between the object and its surroundings, provided that the temperature difference is not too large. B using calculus, the following model can be deduced from this law. t NEWTON S LAW OF COOLING If D 0 is the initial temperature difference between an object and its surroundings, and if its surroundings have temperature T s, then the temperature of the object at time t is modeled b the function Tt2 T s D 0 e kt where k is a positive constant that depends on the tpe of object. EXAMPLE 7 Newton s Law of Cooling A cup of coffee has a temperature of 200 F and is placed in a room that has a temperature of 70 F. After 0 min the temperature of the coffee is 50 F. (a) Find a function that models the temperature of the coffee at time t. (b) Find the temperature of the coffee after 5 min.

47 SECTION 4.6 Modeling with Eponential and Logarithmic Functions 347 (c) When will the coffee have cooled to 00 F? (d) Illustrate b drawing a graph of the temperature function. SOLUTION (a) The temperature of the room is T s 70 F, and the initial temperature difference is So b Newton s Law of Cooling, the temperature after t minutes is modeled b the function We need to find the constant k associated with this cup of coffee. To do this, we use the fact that when t 0, the temperature is T So we have 70 30e 0k 50 D F 30e 0k 80 Tt e kt T s + D 0 e kt = T(t) Subtract 70 e 0k 8 3 0k ln 8 3 Divide b 30 Take ln of each side T ( F) 200 T=70+30e _ t k 0 ln 8 3 k Solve for k Calculator Substituting this value of k into the epression for Tt2, we get Tt e t (b) We use the function that we found in part (a) with t 5. T e F (c) We use the function that we found in part (a) with Tt2 00 and solve the resulting eponential equation for t e t 00 T s + D 0 e kt = T(t) 30e t 30 Subtract 70 e t 3 3 Divide b T= t (min) FIGURE 7 Temperature of coffee after t minutes t ln 3 3 Take ln of each side t ln 3 3 Solve for t t 30.2 Calculator The coffee will have cooled to 00 F after about half an hour. (d) The graph of the temperature function is sketched in Figure 7. Notice that the line t 70 is a horizontal asmptote. (Wh?) NOW TRY EXERCISE 25 Logarithmic Scales When a phsical quantit varies over a ver large range, it is often convenient to take its logarithm in order to have a more manageable set of numbers. We discuss three such situations: the ph scale, which measures acidit; the Richter scale, which measures the

48 348 CHAPTER 4 Eponential and Logarithmic Functions ph for Some Common Substances Substance ph Milk of magnesia 0.5 Seawater Human blood Crackers Homin Cow s milk Spinach Tomatoes Oranges Apples Limes Batter acid.0 Largest Earthquakes Location Date Magnitude Chile Alaska Sumatra Alaska Kamchatka Chile Ecuador Alaska Sumatra Tibet Kamchatka Indonesia Kuril Islands intensit of earthquakes; and the decibel scale, which measures the loudness of sounds. Other quantities that are measured on logarithmic scales are light intensit, information capacit, and radiation. The ph Scale Chemists measured the acidit of a solution b giving its hdrogen ion concentration until Søren Peter Lauritz Sørensen, in 909, proposed a more convenient measure. He defined where 3H 4 is the concentration of hdrogen ions measured in moles per liter (M). He did this to avoid ver small numbers and negative eponents. For instance, if 3H M, then Solutions with a ph of 7 are defined as neutral, those with ph 7 are acidic, and those with ph 7 are basic. Notice that when the ph increases b one unit, 3H 4 decreases b a factor of 0. EXAMPLE 8 ph Scale and Hdrogen Ion Concentration (a) The hdrogen ion concentration of a sample of human blood was measured to be 3H M. Find the ph and classif the blood as acidic or basic. (b) The most acidic rainfall ever measured occurred in Scotland in 974; its ph was 2.4. Find the hdrogen ion concentration. SOLUTION (a) A calculator gives Since this is greater than 7, the blood is basic. (b) To find the hdrogen ion concentration, we need to solve for 3H 4 in the logarithmic equation So we write it in eponential form. In this case ph 2.4, so NOW TRY EXERCISE 29 ph log3h 4 ph log3h 4 log log3h 4 ph 3H 4 0 ph 3H M The Richter Scale In 935 the American geologist Charles Richter ( ) defined the magnitude M of an earthquake to be M log I S where I is the intensit of the earthquake (measured b the amplitude of a seismograph reading taken 00 km from the epicenter of the earthquake) and S is the intensit of a standard earthquake (whose amplitude is micron 0 4 cm). The magnitude of a standard earthquake is M log S log 0 S ph log

49 SECTION 4.6 Modeling with Eponential and Logarithmic Functions 349 Richter studied man earthquakes that occurred between 900 and 950. The largest had magnitude 8.9 on the Richter scale, and the smallest had magnitude 0. This corresponds to a ratio of intensities of 800,000,000, so the Richter scale provides more manageable numbers to work with. For instance, an earthquake of magnitude 6 is ten times stronger than an earthquake of magnitude 5. EXAMPLE 9 Magnitude of Earthquakes The 906 earthquake in San Francisco had an estimated magnitude of 8.3 on the Richter scale. In the same ear a powerful earthquake occurred on the Colombia-Ecuador border that was four times as intense. What was the magnitude of the Colombia-Ecuador earthquake on the Richter scale? SOLUTION If I is the intensit of the San Francisco earthquake, then from the definition of magnitude we have M log I 8.3 S The intensit of the Colombia-Ecuador earthquake was 4I, so its magnitude was NOW TRY EXERCISE 35 M log 4I S log 4 log I log S EXAMPLE 0 Intensit of Earthquakes The 989 Loma Prieta earthquake that shook San Francisco had a magnitude of 7. on the Richter scale. How man times more intense was the 906 earthquake (see Eample 9) than the 989 event? SOLUTION If I and I 2 are the intensities of the 906 and 989 earthquakes, then we are required to find I /I 2. To relate this to the definition of magnitude, we divide the numerator and denominator b S. Roger Ressmeer/CORBIS Therefore, log I log I /S I 2 I 2 /S log I S log I 2 S Divide numerator and denominator b S Law 2 of logarithms Definition of earthquake magnitude The 906 earthquake was about 6 times as intense as the 989 earthquake. NOW TRY EXERCISE 37 I I 2 0 logi /I The Decibel Scale The ear is sensitive to an etremel wide range of sound intensities. We take as a reference intensit I W/m 2 (watts per square meter) at a frequenc of 000 hertz, which measures a sound that is just barel audible (the threshold of hearing). The pschological sensation of loudness varies with the logarithm of the intensit (the Weber- Fechner Law), so the intensit level B, measured in decibels (db), is defined as B 0 log I I 0

50 350 CHAPTER 4 Eponential and Logarithmic Functions The intensit level of the barel audible reference sound is The intensit levels of sounds that we can hear var from ver loud to ver soft. Here are some eamples of the decibel levels of commonl heard sounds. Source of sound B db2 2 Jet takeoff 40 Jackhammer 30 Rock concert 20 Subwa 00 Heav traffic 80 Ordinar traffic 70 Normal conversation 50 Whisper 30 Rustling leaves 0 20 Threshold of hearing 0 EXAMPLE Sound Intensit of a Jet Takeoff Find the decibel intensit level of a jet engine during takeoff if the intensit was measured at 00 W/m 2. SOLUTION From the definition of intensit level we see that Thus, the intensit level is 40 db. NOW TRY EXERCISE 4 B 0 log I 0 I 0 0 log 0 db B 0 log I I 0 0 log log db The table in the margin lists decibel intensit levels for some common sounds ranging from the threshold of human hearing to the jet takeoff of Eample. The threshold of pain is about 20 db. 4.6 EXERCISES APPLICATIONS 6 These eercises use the population growth model.. Bacteria Culture A certain culture of the bacterium Streptococcus A initiall has 0 bacteria and is observed to double ever.5 hours. (a) Find an eponential model nt2 n 0 2 t/a for the number of bacteria in the culture after t hours. (b) Estimate the number of bacteria after 35 hours. (c) When will the bacteria count reach 0,000? Image copright Sebastian Kaulitzki Used under license from Shutterstock.com Streptococcus A 2,000 magnification2 2. Bacteria Culture A certain culture of the bacterium Rhodobacter sphaeroides initiall has 25 bacteria and is observed to double ever 5 hours. (a) Find an eponential model nt2 n 0 2 t/a for the number of bacteria in the culture after t hours. (b) Estimate the number of bacteria after 8 hours. (c) After how man hours will the bacteria count reach million? 3. Squirrel Population A gre squirrel population was introduced in a certain count of Great Britain 30 ears ago. Biologists observe that the population doubles ever 6 ears, and now the population is 00,000. (a) What was the initial size of the squirrel population? (b) Estimate the squirrel population 0 ears from now. (c) Sketch a graph of the squirrel population. 4. Bird Population A certain species of bird was introduced in a certain count 25 ears ago. Biologists observe that the population doubles ever 0 ears, and now the population is 3,000. (a) What was the initial size of the bird population? (b) Estimate the bird population 5 ears from now. (c) Sketch a graph of the bird population. 5. Fo Population The fo population in a certain region has a relative growth rate of 8% per ear. It is estimated that the population in 2005 was 8,000. (a) Find a function nt2 n 0 e rt that models the population t ears after (b) Use the function from part (a) to estimate the fo population in the ear 203. (c) Sketch a graph of the fo population function for the ears Fish Population The population of a certain species of fish has a relative growth rate of.2% per ear. It is estimated that the population in 2000 was 2 million. (a) Find an eponential model nt2 n 0 e rt for the population t ears after (b) Estimate the fish population in the ear (c) Sketch a graph of the fish population.

51 SECTION 4.6 Modeling with Eponential and Logarithmic Functions Population of a Countr The population of a countr has a relative growth rate of 3% per ear. The government is tring to reduce the growth rate to 2%. The population in 995 was approimatel 0 million. Find the projected population for the ear 2020 for the following conditions. (a) The relative growth rate remains at 3% per ear. (b) The relative growth rate is reduced to 2% per ear. 8. Bacteria Culture It is observed that a certain bacteria culture has a relative growth rate of 2% per hour, but in the presence of an antibiotic the relative growth rate is reduced to 5% per hour. The initial number of bacteria in the culture is 22. Find the projected population after 24 hours for the following conditions. (a) No antibiotic is present, so the relative growth rate is 2%. (b) An antibiotic is present in the culture, so the relative growth rate is reduced to 5%. 9. Population of a Cit The population of a certain cit was 2,000 in 2006, and the observed doubling time for the population is 8 ears. (a) Find an eponential model nt2 n 0 2 t/a for the population t ears after (b) Find an eponential model nt2 n 0 e rt for the population t ears after (c) Sketch a graph of the population at time t. (d) Estimate when the population will reach 500, Bat Population The bat population in a certain Midwestern count was 350,000 in 2009, and the observed doubling time for the population is 25 ears. (a) Find an eponential model nt2 n 0 2 t/a for the population t ears after (b) Find an eponential model nt2 n 0 e rt for the population t ears after (c) Sketch a graph of the population at time t. (d) Estimate when the population will reach 2 million.. Deer Population The graph shows the deer population in a Pennslvania count between 2003 and Assume that the population grows eponentiall. (a) What was the deer population in 2003? (b) Find a function that models the deer population t ears after (c) What is the projected deer population in 20? (d) In what ear will the deer population reach 00,000? Deer population n(t) 30,000 20,000 0,000 (4, 3,000) Years since Frog Population Some bullfrogs were introduced into a small pond. The graph shows the bullfrog population for the net few ears. Assume that the population grows eponentiall. (a) What was the initial bullfrog population? t (b) Find a function that models the bullfrog population t ears since the bullfrogs were put into the pond. (c) What is the projected bullfrog population after 5 ears? (d) Estimate how long it takes the population to reach 75,000. n Frog 500 population (2, 225) t 3. Bacteria Culture A culture starts with 8600 bacteria. After one hour the count is 0,000. (a) Find a function that models the number of bacteria nt2 after t hours. (b) Find the number of bacteria after 2 hours. (c) After how man hours will the number of bacteria double? 4. Bacteria Culture The count in a culture of bacteria was 400 after 2 hours and 25,600 after 6 hours. (a) What is the relative rate of growth of the bacteria population? Epress our answer as a percentage. (b) What was the initial size of the culture? (c) Find a function that models the number of bacteria nt2 after t hours. (d) Find the number of bacteria after 4.5 hours. (e) When will the number of bacteria be 50,000? 5. Population of California The population of California was million in 990 and million in Assume that the population grows eponentiall. (a) Find a function that models the population t ears after 990. (b) Find the time required for the population to double. (c) Use the function from part (a) to predict the population of California in the ear 200. Look up California s actual population in 200, and compare. 6. World Population The population of the world was 5.7 billion in 995, and the observed relative growth rate was 2% per ear. (a) B what ear will the population have doubled? (b) B what ear will the population have tripled? 7 24 These eercises use the radioactive deca model. 7. Radioactive Radium The half-life of radium-226 is 600 ears. Suppose we have a 22-mg sample. (a) Find a function mt2 m 0 2 t/h that models the mass remaining after t ears. (b) Find a function mt2 m 0 e rt that models the mass remaining after t ears. (c) How much of the sample will remain after 4000 ears? (d) After how long will onl 8 mg of the sample remain? 8. Radioactive Cesium The half-life of cesium-37 is 30 ears. Suppose we have a 0-g sample. (a) Find a function mt2 m 0 2 t/h that models the mass remaining after t ears.

52 352 CHAPTER 4 Eponential and Logarithmic Functions (b) Find a function mt2 m 0 e rt that models the mass remaining after t ears. (c) How much of the sample will remain after 80 ears? (d) After how long will onl 2 g of the sample remain? 9. Radioactive Strontium The half-life of strontium-90 is 28 ears. How long will it take a 50-mg sample to deca to a mass of 32 mg? 20. Radioactive Radium Radium-22 has a half-life of 30 s. How long will it take for 95% of a sample to deca? 2. Finding Half-life If 250 mg of a radioactive element decas to 200 mg in 48 hours, find the half-life of the element. 22. Radioactive Radon After 3 das a sample of radon-222 has decaed to 58% of its original amount. (a) What is the half-life of radon-222? (b) How long will it take the sample to deca to 20% of its original amount? 23. Carbon-4 Dating A wooden artifact from an ancient tomb contains 65% of the carbon-4 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-4 is 5730 ears.) 24. Carbon-4 Dating The burial cloth of an Egptian mumm is estimated to contain 59% of the carbon-4 it contained originall. How long ago was the mumm buried? (The half-life of carbon-4 is 5730 ears.) (a) If the temperature of the turke is 50 F after half an hour, what is its temperature after 45 min? (b) When will the turke cool to 00 F? 28. Boiling Water A kettle full of water is brought to a boil in a room with temperature 20 C. After 5 min the temperature of the water has decreased from 00 C to 75 C. Find the temperature after another 0 min. Illustrate b graphing the temperature function These eercises deal with logarithmic scales. 29. Finding ph The hdrogen ion concentration of a sample of each substance is given. Calculate the ph of the substance. (a) Lemon juice: 3H M (b) Tomato juice: 3H M (c) Seawater: 3H M 30. Finding ph An unknown substance has a hdrogen ion concentration of 3H M. Find the ph and classif the substance as acidic or basic. 3. Ion Concentration The ph reading of a sample of each substance is given. Calculate the hdrogen ion concentration of the substance. (a) Vinegar: ph 3.0 (b) Milk: ph Ion Concentration The ph reading of a glass of liquid is given. Find the hdrogen ion concentration of the liquid. (a) Beer: ph 4.6 (b) Water: ph Finding ph The hdrogen ion concentrations in cheeses range from M to M. Find the corresponding range of ph readings These eercises use Newton s Law of Cooling. 25. Cooling Soup A hot bowl of soup is served at a dinner part. It starts to cool according to Newton s Law of Cooling, so its temperature at time t is given b Tt e 0.05t where t is measured in minutes and T is measured in F. (a) What is the initial temperature of the soup? (b) What is the temperature after 0 min? (c) After how long will the temperature be 00 F? 26. Time of Death Newton s Law of Cooling is used in homicide investigations to determine the time of death. The normal bod temperature is 98.6 F. Immediatel following death, the bod begins to cool. It has been determined eperimentall that the constant in Newton s Law of Cooling is approimatel k 0.947, assuming that time is measured in hours. Suppose that the temperature of the surroundings is 60 F. (a) Find a function Tt2 that models the temperature t hours after death. (b) If the temperature of the bod is now 72 F, how long ago was the time of death? 27. Cooling Turke A roasted turke is taken from an oven when its temperature has reached 85 F and is placed on a table in a room where the temperature is 75 F. 34. Ion Concentration in Wine The ph readings for wines var from 2.8 to 3.8. Find the corresponding range of hdrogen ion concentrations. 35. Earthquake Magnitudes If one earthquake is 20 times as intense as another, how much larger is its magnitude on the Richter scale? 36. Earthquake Magnitudes The 906 earthquake in San Francisco had a magnitude of 8.3 on the Richter scale. At the same time in Japan an earthquake with magnitude 4.9 caused onl minor damage. How man times more intense was the San Francisco earthquake than the Japanese earthquake? 37. Earthquake Magnitudes The Alaska earthquake of 964 had a magnitude of 8.6 on the Richter scale. How man times more intense was this than the 906 San Francisco earthquake? (See Eercise 36.)

53 CHAPTER 4 Review Earthquake Magnitudes The Northridge, California, earthquake of 994 had a magnitude of 6.8 on the Richter scale. A ear later, a 7.2-magnitude earthquake struck Kobe, Japan. How man times more intense was the Kobe earthquake than the Northridge earthquake? 39. Earthquake Magnitudes The 985 Meico Cit earthquake had a magnitude of 8. on the Richter scale. The 976 earthquake in Tangshan, China, was.26 times as intense. What was the magnitude of the Tangshan earthquake? 40. Subwa Noise The intensit of the sound of a subwa train was measured at 98 db. Find the intensit in W/m Traffic Noise The intensit of the sound of traffic at a bus intersection was measured at W/m 2. Find the intensit level in decibels. 42. Comparing Decibel Levels The noise from a power mower was measured at 06 db. The noise level at a rock concert was measured at 20 db. Find the ratio of the intensit of the rock music to that of the power mower. 43. Inverse Square Law for Sound A law of phsics states that the intensit of sound is inversel proportional to the square of the distance d from the source: I k/d 2. (a) Use this model and the equation B 0 log I I 0 (described in this section) to show that the decibel levels B and B 2 at distances d and d 2 from a sound source are related b the equation B 2 B 20 log d d 2 (b) The intensit level at a rock concert is 20 db at a distance 2 m from the speakers. Find the intensit level at a distance of 0 m. CHAPTER 4 REVIEW CONCEPT CHECK. (a) Write an equation that defines the eponential function with base a. (b) What is the domain of this function? (c) What is the range of this function? (d) Sketch the general shape of the graph of the eponential function for each case. (i) a (ii) 0 a 2. If is large, which function grows faster, 2 or 2? 3. (a) How is the number e defined? (b) What is the natural eponential function? 4. (a) How is the logarithmic function log a defined? (b) What is the domain of this function? (c) What is the range of this function? (d) Sketch the general shape of the graph of the function log a if a. (e) What is the natural logarithm? (f) What is the common logarithm? 5. State the three Laws of Logarithms. 6. State the Change of Base Formula. 7. (a) How do ou solve an eponential equation? (b) How do ou solve a logarithmic equation? 8. Suppose an amount P is invested at an interest rate r and A is the amount after t ears. (a) Write an epression for A if the interest is compounded n times per ear. (b) Write an epression for A if the interest is compounded continuousl. 9. The initial size of a population is n 0 and the population grows eponentiall. (a) Write an epression for the population in terms of the doubling time a. (b) Write an epression for the population in terms of the relative growth rate r. 0. (a) What is the half-life of a radioactive substance? (b) If a radioactive substance has initial mass m 0 and half-life h, write an epression for the mass mt2 remaining at time t.. What does Newton s Law of Cooling sa? 2. What do the ph scale, the Richter scale, and the decibel scale have in common? What do the measure? EXERCISES 4 Use a calculator to find the indicated values of the eponential function, correct to three decimal places.. f2 5 ; f.52, f 222, f f2 3 # 2 ; f 2.22, f 272, f g2 4 # A 2 3B 2 ; g 0.72, ge2, gp2 4. g2 7 4e ; g 22, g 232, g Sketch the graph of the function. State the domain, range, and asmptote f f g g2 5 5

54 354 CHAPTER 4 Eponential and Logarithmic Functions f2 log g2 log 2. f2 2 log 2 2. f2 3 log F2 e 4. G 2 2 e 5. g2 2 ln 6. g2 ln Find the domain of the function f2 0 2 log 22 g2 log h2 ln k2 ln 2 24 Write the equation in eponential form. 2. log log log 24. ln c Write the equation in logarithmic form / e k m Evaluate the epression without using a calculator. 29. log log log log lne log log 3 A 27 B log log 23 e 2ln7 39. log 25 log log log log log e 2 2e 2 8e log log log 2 log log 8 52 log ln Use a calculator to find the solution of the equation, rounded to si decimal places / e 5k 0, Draw a graph of the function and use it to determine the asmptotes and the local maimum and minimum values. 73. e / log ln Find the solutions of the equation, rounded to two decimal places log e Solve the inequalit graphicall. 79. ln e Use a graph of f2 e 3e 4 to find, approimatel, the intervals on which f is increasing and on which f is decreasing. 82. Find an equation of the line shown in the figure. e 3/ log 8 6 log 8 3 log log log 0 00 =ln Epand the logarithmic epression. 45. logab 2 C log e a ln log a B 2 2 b log 2 5 a /2 b 50. ln a b Combine into a single logarithm. 5. log 6 4 log log log log log log log 5 2 log log log 22 log 22 2 log ln 42 5 ln Solve the equation. Find the eact solution if possible; otherwise, use a calculator to approimate to two decimals Use the Change of Base Formula to evaluate the logarithm, rounded to si decimal places. 83. log log log log Which is larger, log or log 5 620? 88. Find the inverse of the function f2 2 3, and state its domain and range. 89. If $2,000 is invested at an interest rate of 0% per ear, find the amount of the investment at the end of 3 ears for each compounding method. (a) Semiannuall (b) Monthl (c) Dail (d) Continuousl 90. A sum of $5000 is invested at an interest rate of % per ear, compounded semiannuall. (a) Find the amount of the investment after ears

55 CHAPTER 4 Review 355 (b) After what period of time will the investment amount to $7000? (c) If interest were compounded continousl instead of semiannuall, how long would it take for the amount to grow to $7000? 9. A mone market account pas 5.2% annual interest, compounded dail. If $00,000 is invested in this account, how long will it take for the account to accumulate $0,000 in interest? 92. A retirement savings plan pas 4.5% interest, compounded continuousl. How long will it take for an investment in this plan to double? Determine the annual percentage ield (APY) for the given nominal annual interest rate and compounding frequenc %; dail %; monthl 95. The stra-cat population in a small town grows eponentiall. In 999 the town had 30 stra cats, and the relative growth rate was 5% per ear. (a) Find a function that models the stra-cat population nt2 after t ears. (b) Find the projected population after 4 ears. (c) Find the number of ears required for the stra-cat population to reach A culture contains 0,000 bacteria initiall. After an hour the bacteria count is 25,000. (a) Find the doubling period. (b) Find the number of bacteria after 3 hours. 97. Uranium-234 has a half-life of ears. (a) Find the amount remaining from a 0-mg sample after a thousand ears. (b) How long will it take this sample to decompose until its mass is 7 mg? 98. A sample of bismuth-20 decaed to 33% of its original mass after 8 das. (a) Find the half-life of this element. (b) Find the mass remaining after 2 das. 99. The half-life of radium-226 is 590 ears. (a) If a sample has a mass of 50 mg, find a function that models the mass that remains after t ears. (b) Find the mass that will remain after 000 ears. (c) After how man ears will onl 50 mg remain? 00. The half-life of palladium-00 is 4 das. After 20 das a sample has been reduced to a mass of g. (a) What was the initial mass of the sample? (b) Find a function that models the mass remaining after t das. (c) What is the mass after 3 das? (d) After how man das will onl 0.5 g remain? 0. The graph shows the population of a rare species of bird, where t represents ears since 999 and nt2 is measured in thousands. (a) Find a function that models the bird population at time t in the form nt2 n 0 e rt. (b) What is the bird population epected to be in the ear 200? Bird population n(t) (5, 3200) t Years since A car engine runs at a temperature of 90 F. When the engine is turned off, it cools according to Newton s Law of Cooling with constant k 0.034, where the time is measured in minutes. Find the time needed for the engine to cool to 90 F if the surrounding temperature is 60 F. 03. The hdrogen ion concentration of fresh egg whites was measured as 3H M Find the ph, and classif the substance as acidic or basic. 04. The ph of lime juice is.9. Find the hdrogen ion concentration. 05. If one earthquake has magnitude 6.5 on the Richter scale, what is the magnitude of another quake that is 35 times as intense? 06. The drilling of a jackhammer was measured at 32 db. The sound of whispering was measured at 28 db. Find the ratio of the intensit of the drilling to that of the whispering.

56 CHAPTER 4 TEST. Sketch the graph of each function, and state its domain, range, and asmptote. Show the - and -intercepts on the graph. (a) f2 2 4 (b) g2 log (a) Write the equation in logarithmic form. (b) Write the equation ln A 3 in eponential form. 3. Find the eact value of each epression. (a) log 36 0 (b) ln e 3 (c) log 3 27 (d) log 2 80 log 2 0 (e) log 8 4 (f) log 6 4 log Use the Laws of Logarithms to epand the epression: 3 2 log B Combine into a single logarithm: ln 2 ln ln Find the solution of the equation, correct to two decimal places. (a) 2 0 (b) 5 ln3 2 4 (c) (d) log 2 22 log The initial size of a culture of bacteria is 000. After one hour the bacteria count is (a) Find a function that models the population after t hours. (b) Find the population after.5 hours. (c) When will the population reach 5,000? (d) Sketch the graph of the population function. 8. Suppose that $2,000 is invested in a savings account paing 5.6% interest per ear. (a) Write the formula for the amount in the account after t ears if interest is compounded monthl. (b) Find the amount in the account after 3 ears if interest is compounded dail. (c) How long will it take for the amount in the account to grow to $20,000 if interest is compounded semiannuall? 9. The half-life of krpton-9 ( 9 Kr) is 0 seconds. At time t = 0 a heav canister contains 3 g of this radioactive gas. (a) Find a function that models the amount A(t) of 9 Kr remaining in the canister after t seconds. (b) How much 9 Kr remains after one minute? (c) When will the amount of 9 Kr remaining be reduced to mg ( microgram, or 0 6 g)? 0. An earthquake measuring 6.4 on the Richter scale struck Japan in Jul 2007, causing etensive damage. Earlier that ear, a minor earthquake measuring 3. on the Richter scale was felt in parts of Pennslvania. How man times more intense was the Japanese earthquake than the Pennslvania earthquake? 356

57 FOCUS ON MODELING Fitting Eponential and Power Curves to Data In a previous Focus on Modeling (page 296) we learned that the shape of a scatter plot helps us to choose the tpe of curve to use in modeling data. The first plot in Figure fairl screams for a line to be fitted through it, and the second one points to a cubic polnomial. For the third plot it is tempting to fit a second-degree polnomial. But what if an eponential curve fits better? How do we decide this? In this section we learn how to fit eponential and power curves to data and how to decide which tpe of curve fits the data better. We also learn that for scatter plots like those in the last two plots in Figure, the data can be modeled b logarithmic or logistic functions. FIGURE Modeling with Eponential Functions If a scatter plot shows that the data increase rapidl, we might want to model the data using an eponential model, that is, a function of the form f2 Ce k where C and k are constants. In the first eample we model world population b an eponential model. Recall from Section 4.6 that population tends to increase eponentiall. TABLE World population Year (t) World population (P in millions) EXAMPLE An Eponential Model for World Population Table gives the population of the world in the 20th centur. (a) Draw a scatter plot, and note that a linear model is not appropriate. (b) Find an eponential function that models population growth. (c) Draw a graph of the function that ou found together with the scatter plot. How well does the model fit the data? (d) Use the model that ou found to predict world population in the ear SOLUTION (a) The scatter plot is shown in Figure 2. The plotted points do not appear to lie along a straight line, so a linear model is not appropriate FIGURE 2 Scatter plot of world population 357

58 Chabruken / The Image Bank /Gett Images 358 Focus on Modeling The population of the world increases eponentiall. (b) Using a graphing calculator and the EpReg command (see Figure 3(a)), we get the eponential model This is a model of the form Cb t. To convert this to the form Ce kt, we use the properties of eponentials and logarithms as follows: Thus, we can write the model as Pt # t t ln.03786t e t ln e e t A = e lna ln A B = B ln A Pt e t ln (c) From the graph in Figure 3(b) we see that the model appears to fit the data fairl well. The period of relativel slow population growth is eplained b the depression of the 930s and the two world wars (a) (b) 2000 FIGURE 3 Eponential model for world population (d) The model predicts that the world population in 2020 will be P e 7,405,400,000 Modeling with Power Functions If the scatter plot of the data we are studing resembles the graph of a 2, a.32, or some other power function, then we seek a power model, that is, a function of the form f2 a n Sun Mercur Mars Venus Earth Saturn Jupiter where a is a positive constant and n is an real number. In the net eample we seek a power model for some astronomical data. In astronom, distance in the solar sstem is often measured in astronomical units. An astronomical unit (AU) is the mean distance from the earth to the sun. The period of a planet is the time it takes the planet to make a complete revolution around the sun (measured in earth ears). In this eample we derive the remarkable relationship, first discovered b Johannes Kepler (see page 754), between the mean distance of a planet from the sun and its period. EXAMPLE 2 A Power Model for Planetar Periods Table 2 gives the mean distance d of each planet from the sun in astronomical units and its period T in ears.

59 Fitting Eponential and Power Curves to Data 359 TABLE 2 Distances and periods of the planets Planet d T Mercur Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto (a) Sketch a scatter plot. Is a linear model appropriate? (b) Find a power function that models the data. (c) Draw a graph of the function ou found and the scatter plot on the same graph. How well does the model fit the data? (d) Use the model that ou found to calculate the period of an asteroid whose mean distance from the sun is 5 AU. SOLUTION (a) The scatter plot shown in Figure 4 indicates that the plotted points do not lie along a straight line, so a linear model is not appropriate. 260 FIGURE 4 Scatter plot of planetar data 0 45 (b) Using a graphing calculator and the PwrReg command (see Figure 5(a)), we get the power model T d If we round both the coefficient and the eponent to three significant figures, we can write the model as T d.5 This is the relationship discovered b Kepler (see page 754). Sir Isaac Newton (page 852) later used his Law of Gravit to derive this relationship theoreticall, thereb providing strong scientific evidence that the Law of Gravit must be true. (c) The graph is shown in Figure 5(b). The model appears to fit the data ver well. 260 FIGURE 5 Power model for planetar data (a) 0 (b) 45 (d) In this case d 5 AU, so our model gives T # The period of the asteroid is about.2 ears. Linearizing Data We have used the shape of a scatter plot to decide which tpe of model to use: linear, eponential, or power. This works well if the data points lie on a straight line. But it s difficult to distinguish a scatter plot that is eponential from one that requires a power model. So to help decide which model to use, we can linearize the data, that is, appl a function that straightens the scatter plot. The inverse of the linearizing function is then

60 360 Focus on Modeling an appropriate model. We now describe how to linearize data that can be modeled b eponential or power functions. TABLE 3 World population data Population P t (in millions) ln P TABLE 4 Log-log table ln d ln T Linearizing eponential data If we suspect that the data points, 2 lie on an eponential curve Ce k, then the points, ln 2 should lie on a straight line. We can see this from the following calculations: ln ln Ce k Assume that = Ce k and take ln ln e k ln C Propert of ln k ln C Propert of ln To see that ln is a linear function of, let Y ln and A ln C; then Y k A We appl this technique to the world population data t, P2 to obtain the points t, ln P2 in Table 3. The scatter plot of t, ln P2 in Figure 6, called a semi-log plot, shows that the linearized data lie approimatel on a straight line, so an eponential model should be appropriate. FIGURE 6 Semi-log plot of data in Table Linearizing power data If we suspect that the data points, 2 lie on a power curve a n, then the points ln, ln 2 should be on a straight line. We can see this from the following calculations: ln ln a n Assume that = a n and take ln ln a ln n Propert of ln ln a n ln Propert of ln To see that ln is a linear function of ln, let Y ln, X ln, and A ln a; then Y nx A We appl this technique to the planetar data d, T2 in Table 2 to obtain the points ln d, ln T2 in Table 4. The scatter plot of ln d, ln T2 in Figure 7, called a log-log plot, shows that the data lie on a straight line, so a power model seems appropriate. FIGURE 7 Log-log plot of data in Table 4 _2 6 _

61 An Eponential or Power Model? Fitting Eponential and Power Curves to Data 36 Suppose that a scatter plot of the data points, 2 shows a rapid increase. Should we use an eponential function or a power function to model the data? To help us decide, we draw two scatter plots: one for the points, ln 2 and the other for the points ln, ln 2. If the first scatter plot appears to lie along a line, then an eponential model is appropriate. If the second plot appears to lie along a line, then a power model is appropriate. TABLE EXAMPLE 3 An Eponential or Power Model? Data points, 2 are shown in Table 5. (a) Draw a scatter plot of the data. (b) Draw scatter plots of, ln 2 and ln, ln 2. (c) Is an eponential function or a power function appropriate for modeling this data? (d) Find an appropriate function to model the data. SOLUTION (a) The scatter plot of the data is shown in Figure FIGURE 8 0 TABLE ln ln FIGURE (b) We use the values from Table 6 to graph the scatter plots in Figures 9 and FIGURE 9 Semi-log plot (c) The scatter plot of, ln 2 in Figure 9 does not appear to be linear, so an eponential model is not appropriate. On the other hand, the scatter plot of ln, ln 2 in Figure 0 is ver nearl linear, so a power model is appropriate. (d) Using the PwrReg command on a graphing calculator, we find that the power function that best fits the data point is FIGURE 0 Log-log plot The graph of this function and the original data points are shown in Figure. Before graphing calculators and statistical software became common, eponential and power models for data were often constructed b first finding a linear model for the linearized data. Then the model for the actual data was found b taking eponentials. For instance, if we find that ln A ln B, then b taking eponentials we get the model e B e A ln,or C A (where C e B ). Special graphing paper called log paper or log-log paper was used to facilitate this process

62 362 Focus on Modeling Modeling with Logistic Functions A logistic growth model is a function of the form ft2 c ae bt where a, b, and c are positive constants. Logistic functions are used to model populations where the growth is constrained b available resources. (See Eercises of Section 4.2.) TABLE 7 Week Catfish EXAMPLE 4 Stocking a Pond with Catfish Much of the fish that is sold in supermarkets toda is raised on commercial fish farms, not caught in the wild. A pond on one such farm is initiall stocked with 000 catfish, and the fish population is then sampled at 5-week intervals to estimate its size. The population data are given in Table 7. (a) Find an appropriate model for the data. (b) Make a scatter plot of the data and graph the model that ou found in part (a) on the scatter plot. (c) How does the model predict that the fish population will change with time? SOLUTION (a) Since the catfish population is restricted b its habitat (the pond), a logistic model is appropriate. Using the Logistic command on a calculator (see Figure 2(a)), we find the following model for the catfish population Pt2: Pt e 0.052t 9000 FIGURE 2 (a) 0 (b) Catfish population = P(t) 80 (b) The scatter plot and the logistic curve are shown in Figure 2(b). (c) From the graph of P in Figure 2(b) we see that the catfish population increases rapidl until about t 80 weeks. Then growth slows down, and at about t 20 weeks the population levels off and remains more or less constant at slightl over The behavior that is ehibited b the catfish population in Eample 4 is tpical of logistic growth. After a rapid growth phase, the population approaches a constant level called the carring capacit of the environment. This occurs because as t q, we have e bt 0 (see Section 4.2), and so c Pt2 ae bt c 0 c Thus, the carring capacit is c.

63 PROBLEMS. U.S. Population The U.S. Constitution requires a census ever 0 ears. The census data for are given in the table. (a) Make a scatter plot of the data. (b) Use a calculator to find an eponential model for the data. (c) Use our model to predict the population at the 200 census. (d) Use our model to estimate the population in 965. Fitting Eponential and Power Curves to Data 363 (e) Compare our answers from parts (c) and (d) to the values in the table. Do ou think an eponential model is appropriate for these data? Population Population Population Year (in millions) Year (in millions) Year (in millions) Time (s) Distance (m) A Falling Ball In a phsics eperiment a lead ball is dropped from a height of 5 m. The students record the distance the ball has fallen ever one-tenth of a second. (This can be done b using a camera and a strobe light.) (a) Make a scatter plot of the data. (b) Use a calculator to find a power model. (c) Use our model to predict how far a dropped ball would fall in 3 s. 3. Health-Care Ependitures The U.S. health-care ependitures for are given in the table below, and a scatter plot of the data is shown in the figure. (a) Does the scatter plot shown suggest an eponential model? (b) Make a table of the values t, ln E2 and a scatter plot. Does the scatter plot appear to be linear? (c) Find the regression line for the data in part (b). (d) Use the results of part (c) to find an eponential model for the growth of health-care ependitures. (e) Use our model to predict the total health-care ependitures in Year Health-care ependitures (in billions of dollars) U.S. health-care ependitures (in billions of dollars) E t Year

64 364 Focus on Modeling Time (h) Amount of 3 I g Half-Life of Radioactive Iodine A student is tring to determine the half-life of radioactive iodine-3. He measures the amount of iodine-3 in a sample solution ever 8 hours. His data are shown in the table in the margin. (a) Make a scatter plot of the data. (b) Use a calculator to find an eponential model. (c) Use our model to find the half-life of iodine The Beer-Lambert Law As sunlight passes through the waters of lakes and oceans, the light is absorbed, and the deeper it penetrates, the more its intensit diminishes. The light intensit I at depth is given b the Beer-Lambert Law: I I 0 e k where I 0 is the light intensit at the surface and k is a constant that depends on the murkiness of the water (see page 336). A biologist uses a photometer to investigate light penetration in a northern lake, obtaining the data in the table. (a) Use a graphing calculator to find an eponential function of the form given b the Beer- Lambert Law to model these data. What is the light intensit I 0 at the surface on this da, and what is the murkiness constant k for this lake? [Hint: If our calculator gives ou a function of the form I ab, convert this to the form ou want using the identities b e ln b 2 e ln b. See Eample (b).] (b) Make a scatter plot of the data, and graph the function that ou found in part (a) on our scatter plot. (c) If the light intensit drops below 0.5 lumen (lm), a certain species of algae can t survive because photosnthesis is impossible. Use our model from part (a) to determine the depth below which there is insufficient light to support this algae. Depth Light intensit Depth Light intensit (ft) (lm) (ft) (lm) Light intensit decreases eponentiall with depth Eperimenting with Forgetting Curves Ever one of us is all too familiar with the phenomenon of forgetting. Facts that we clearl understood at the time we first learned them sometimes fade from our memor b the time the final eam rolls around. Pschologists have proposed several was to model this process. One such model is Ebbinghaus Law of Forgetting, described on page 327. Other models use eponential or logarithmic functions. To develop her own model, a pschologist performs an eperiment on a group of volunteers b asking them to memorize a list of 00 related words. She then tests how man of these words the can recall after various periods of time. The average results for the group are shown in the table. (a) Use a graphing calculator to find a power function of the form at b that models the average number of words that the volunteers remember after t hours. Then find an eponential function of the form ab t to model the data. (b) Make a scatter plot of the data, and graph both the functions that ou found in part (a) on our scatter plot. (c) Which of the two functions seems to provide the better model? Time Words recalled 5 min 64.3 h h 37.3 da das das das 22.9

65 Fitting Eponential and Power Curves to Data 365 Image copright ARENA Creative. Used under license from Shutterstock.com The number of different bat species in a cave is related to the size of the cave b a power function. 7. Modeling the Species-Area Relation The table gives the areas of several caves in central Meico and the number of bat species that live in each cave.* (a) Find a power function that models the data. (b) Draw a graph of the function ou found in part (a) and a scatter plot of the data on the same graph. Does the model fit the data well? (c) The cave called El Sapo near Puebla, Meico, has a surface area of A 205 m 2. Use the model to estimate the number of bat species ou would epect to find in that cave. Cave Area m 2 2 Number of species La Escondida 8 El Escorpion 9 El Tigre 58 Mision Imposible 60 2 San Martin 28 5 El Arenal 87 4 La Ciudad Virgen Auto Ehaust Emissions A stud b the U.S. Office of Science and Technolog in 972 estimated the cost of reducing automobile emissions b certain percentages. Find an eponential model that captures the diminishing returns trend of these data shown in the table below. Reduction in Cost per emissions (%) car ($) Eponential or Power Model? Data points, 2 are shown in the table. (a) Draw a scatter plot of the data. (b) Draw scatter plots of, ln 2 and ln, ln 2. (c) Which is more appropriate for modeling this data: an eponential function or a power function? (d) Find an appropriate function to model the data *A. K. Brunet and R. A. Medallin, The Species-Area Relationship in Bat Assemblages of Tropical Caves. Journal of Mammalog, 82(4):4 22, 200.