314 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS Section 4-7 Exponential and Logarithmic Equations Exponential Equations Logarithmic Equations Change of Base Equations involving exponential and logarithmic functions, such as 3x and log (x 3) log x 1 are called exponential and logarithmic equations, respectively. Logarithmic properties play a central role in their solution. Of course, a graphing utility can be used to find approximate solutions for many exponential and logarithmic equations. However, there are situations where the algebraic solution is necessary. In this section, we emphasize algebraic solutions and use a graphing utility as a check, when appropriate. Exponential Equations The following examples illustrate the use of logarithmic properties in solving exponential equations. 1 Solving an Exponential Equation Solve 3x for x to four decimal places. How can we get x out of the exponent? Use logs! Since the logarithm function is one-to-one, if two positive quantities are equal, their logs are equal. See Theorem 1 in Section 4-. 3x log 3x log Take the common or natural log of both sides. FIGURE 1 y 1 3x, y. 8 4 (3x ) log log 3x log log x 1 3 log log 1.446 Use log b N p p log b N to get 3x out of the exponent position. Remember: log log log. log To four decimal places. Figure 1 shows a graphical solution that confirms this result. 1 Solve 3 1 x 7 for x to four decimal places.
4-7 Exponential and Logarithmic Equations 31 Compound Interest A certain amount of money P (principal) is invested at an annual rate r compounded annually. The amount of money A in the account after t years, assuming no withdrawals, is given by A P 1 r m n P(1 r) n m 1 for annual compounding. How many years to the nearest year will it take the money to double if it is invested at 6% compounded annually? To find the doubling time, we replace A in A P(1.6) n with P and solve for n. P P(1.6) n 1.6 n Divide both sides by P. FIGURE y 1 1.6 x, y. 4 log log 1.6 n n log 1.6 n log log 1.6 1 years Take the common or natural log of both sides. Note how log properties are used to get n out of the exponent position. To the nearest year. Figure confirms this result. Repeat Example, changing the interest rate to 9% compounded annually. 3 Atmospheric Pressure The atmospheric pressure P, in pounds per square inch, at x miles above sea level is given approximately by P 14.7e.1x At what height will the atmospheric pressure be half the sea-level pressure? Compute the answer to two significant digits. Sea-level pressure is the pressure at x. Thus, P 14.7e 14.7 One-half of sea-level pressure is 14.7/ 7.3. Now our problem is to find x so that P 7.3; that is, we solve 7.3 14.7e.1x for x:
316 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS 7.3 14.7e.1x. e.1x Divide both sides by 14.7 to simplify. FIGURE 3 y 1 14.7e.1x, y 7.3. ln. ln e.1x.1x x ln..1 3.3 miles Since the base is e, take the natural log of both sides. In e 1 To two significant digits. Figure 3 shows that this answer is correct. 3 Using the formula in Example 3, find the altitude in miles so that the atmospheric pressure will be one-eighth that at sea level. Compute the answer to two significant digits. The graph of y ex e x (1) is a curve called a catenary (Fig. 4). A uniform cable suspended between two fixed points is a physical example of such a curve. FIGURE 4 Catenary. 1 y y e x e x x 4 Solving an Exponential Equation Given equation (1), find x for y.. Compute the answer to four decimal places. y ex e x. ex e x e x e x e x e x 1 e x e x 1 Multiply both sides by e x. This is a quadratic in e x.
4-7 Exponential and Logarithmic Equations 317 FIGURE y 1 ex e x, y.. Let u e x, then u u 1 u 4(1)(1) 1 e x 1 ln e x ln 1 x ln 1 1.668, 1.668 Replace u with e x and solve for x. Take the natural log of both sides (both values on the right are positive). log b b x x. Figure confirms the positive solution. Note that the algebraic method also produced exact solutions, an important consideration in certain calculus applications (see Problems 7 6 in Exercise 4-7). 4 Given y (e x e x )/, find x for y 1.. Compute the answer to three decimal places. Explore/Discuss 1 Let y e x 3e x e x (A) Try to find x when y 7 using the method of Example 4. Explain the difficulty that arises. (B) Use a graphing utility to find x when y 7. Logarithmic Equations We now illustrate the solution of several types of logarithmic equations. Solving a Logarithmic Equation Solve log (x 3) log x 1, and check. First use properties of logarithms to express the left side as a single logarithm, then convert to exponential form and solve for x.
318 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS log (x 3) log x 1 log [x(x 3)] 1 x(x 3) 1 1 x 3x 1 (x )(x ) x, Combine left side using log M log N log MN. Change to equivalent exponential form. Write in ax bx c form and solve. Check x : log ( 3) log ( ) is not defined because the domain of the log function is (, ). FIGURE 6 y 1 log (x 3) log x, y 1. x : log ( 3) log log log log ( ) log 1 1 Thus, the only solution to the original equation is x. Remember, answers should be checked in the original equation to see whether any should be discarded. Figure 6 shows the solution at x and also shows that the left side of the equation is not defined at x, the extraneous solution produced by the algebraic method. Solve log (x 1) log x, and check. 6 FIGURE 7 y 1 (ln x), y ln x. 6 Solving a Logarithmic Equation Solve (ln x) ln x. There are no logarithmic properties for simplifying (ln x). However, we can simplify ln x, obtaining an equation involving ln x and (ln x). (ln x) ln x ln x (ln x) ln x (ln x)(ln x ) ln x or ln x x e ln x 1 x e This is a quadratic equation in ln x. Move all nonzero terms to the left and factor. 4 1 Checking that both x 1 and x e are solutions to the original equation is left to you. Figure 7 confirms the solution at e 7.38961.
4-7 Exponential and Logarithmic Equations 319 6 Solve log x (log x). CAUTION Note that (log b x) log b x (log b x) (log b x)(log b x) log b x log b x 7 Earthquake Intensity Recall from Section 4-6 that the magnitude of an earthquake on the Richter scale is given by M 3 log E E Solve for E in terms of the other symbols. M 3 log E E log E E 3M E E 1 3M/ E E 1 3M/ 3 Multiply both sides by. Change to exponential form. 7 Solve the rocket equation from Section 4-6 for W b in terms of the other symbols: v c ln W t W b Change of Base How would you find the logarithm of a positive number to a base other than 1 or e? For example, how would you find log 3.? In Example 8 we evaluate this logarithm using a direct process. Then we develop a change-of-base formula to find such logarithms in general. You may find it easier to remember the process than the formula.
3 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS 8 Evaluating a Base 3 Logarithm Evaluate log 3. to four decimal places. Let y log 3. and proceed as follows: log 3. y. 3 y ln. ln 3 y y ln 3 y ln. ln 3 Change to exponential form. Take the natural log (or common log) of each side. log b M p p log b M Solve for y. Replace y with log 3. from the first step, and use a calculator to evaluate the right side: log 3. ln. ln 3 1.7 8 Evaluate log..37 to four decimal places. To develop a change-of-base formula for arbitrary positive bases, with neither base equal to 1, we proceed as above. Let y log b N, where N and b are positive and b 1. Then log b N y N b y log a N log a b y y log a b y log a N log a b Write in exponential form. Take the log of each side to another positive base a, a 1. log b M p p log b M Solve for y. Replacing y with log b N from the first step, we obtain the chain-of-base formula: log b N log a N log a b In words, this formula states that the logarithm of a number to a given base is the logarithm of that number to a new base divided by the logarithm of the old base to the new base. In practice, we usually choose either e or 1 for the new base so that a calculator can be used to evaluate the necessary logarithms (see Example 8).
4-7 Exponential and Logarithmic Equations 31 Explore/Discuss If b is any positive real number different from 1, the change-of-base formula implies that the function y log b x is a constant multiple of the natural logarithmic function; that is, log b x k ln x for some k. (A) Graph the functions y ln x, y ln x, y. ln x, and y 3 ln x. (B) Write each function of part A in the form y log b x by finding the base b to two decimal places. (C) Is every exponential function y b x a constant multiple of y e x? Explain. Answers to Matched Problems 1. x.63. More than double in 9 years, but not quite double in 8 years 3. 9.9 miles 4. x 1.19. x 6. x 1,1 7. W b W t e v/c 8. 4.7486 EXERCISE 4-7. e x.3 6. e x 1 A Solve Problems 1 1 algebraically and check graphically. Round answers to three significant digits. 1. 1 x.347. 1 x 14.3 3. 1 3x 1 9 4. 1 x 348. e x 3.6 6. e x.14 7. e x 1 4 8. e 3x 3.8 9. x 18 1. 3 x 4 11. x.38 1. 3 x.74 Solve Problems 13 18 exactly. 13. log log x 14. log x log 8 1 1. log x log (x 3) 1 16. log (x 9) log 1x 3 17. log (x 1) log (x 1) 1 18. log (x 1) 1 log (x ) B Solve Problems 19 6 algebraically and check graphically. Round answers to three significant digits. 19. 1. x. 3 1.6 x 1. e 1.4x 13. e.3x 63 3. 13 e.1x 4. 438 e.x Solve Problems 7 38 exactly. 7. log x log log log (x 3) 8. log (6x ) log 3 log log x 9. ln x ln (x 1) ln (x ) 3. ln (x 1) ln (3x 1) ln x 31. log (x 1) 1 log (x 1) 3. 1 log (x ) log (3x 1) 33. (ln x) 3 ln x 4 34. (log x) 3 log x 4 3. ln (ln x) 1 36. log (log x) 1 37. x log x 1x 38. 3 log x 3x In Problems 39 4, (A) Explain the difficulty in solving the equation exactly. (B) Determine the number of solutions by graphing the functions on each side of the equation. 39. e x/ ln x 4. ln (ln x) ln x In Problems 41 4, (A) Explain the difficulty in solving the equation exactly. (B) Use a graphing utility to find all solutions to three decimal places. 41. 3 x 7 x e x 4. e x/4 log x 4 ln x
3 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS Evaluate Problems 43 48 to four decimal places. 43. log 37 44. log 4 3 4. log 8.3 46. log. 439 47. log 3.1483 48. log 1 43.6 C Solve Problems 49 6 for the indicated variable in terms of the remaining symbols. Use the natural log for solving exponential equations. 49. A Pe rt for r (finance). 1.. 3. 4. L 8.8.1 log D for D (astronomy). 6. A P 1 r n nt for t (finance) D 1 log I I for I (sound) t 1 k (ln A ln A ) for A (decay) M 6. log I I for I (astronomy) I E R (1 e Rt/L ) for t (circuitry) S R (1 i)n 1 for n (annuity) i The following combinations of exponential functions define four of six hyperbolic functions, an important class of functions in calculus and higher mathematics. Solve Problems 7 6 for x in terms of y. The results are used to define inverse hyperbolic functions, another important class of functions in calculus and higher mathematics. 7. y ex e x 8. 9. y ex e x 6. e x e x y ex e x y ex e x e x e x In Problems 61 64, use a graphing utility to graph each function. [Hint: Use the change-of-base formula first.] 61. y 3 log ( x) 6. y log 3 (4 x) 63. y log 3 x log x 64. y log 3 x log x In Problems 6 76, use a graphing utility to approximate to two decimal places any solutions of the equation in the interval x 1. None of these equations can be solved exactly using any step-by-step algebraic process. 6. x x 66. 3 x 3x 67. x3 x 1 68. x x 1 69. e x x 7. xe x 1 71. xe x 7. e x x 73. ln x x 74. ln x x 7. ln x e x 76. ln x x APPLICATIONS Solve Problems 77 9 algebraically or graphically, whichever seems more appropriate. 77. Compound Interest. How many years, to the nearest year, will it take a sum of money to double if it is invested at 1% compounded annually? 78. Compound Interest. How many years, to the nearest year, will it take money to quadruple if it is invested at % compounded annually? 79. Compound Interest. At what annual rate compounded continuously will $1, have to be invested to amount to $, in 1 years? Compute the answer to three significant digits. 8. Compound Interest. How many years will it take $, to amount to $8, if it is invested at an annual rate of 9% compounded continuously? Compute the answer to three significant digits. 81. Astronomy. The brightness of stars is expressed in terms of magnitudes on a numerical scale that increases as the brightness decreases. The magnitude m is given by the formula m 6. log L L where L is the light flux of the star and L is the light flux of the dimmest stars visible to the naked eye. (A) What is the magnitude of the dimmest stars visible to the naked eye? (B) How many times brighter is a star of magnitude 1 than a star of magnitude 6? 8. Astronomy. An optical instrument is required to observe stars beyond the sixth magnitude, the limit of ordinary vision. However, even optical instruments have their limitations. The limiting magnitude L of any optical telescope with lens diameter D, in inches, is given by L 8.8.1 log D (A) Find the limiting magnitude for a homemade 6-inch reflecting telescope. (B) Find the diameter of a lens that would have a limiting magnitude of.6. Compute answers to three significant digits. 83. World Population. A mathematical model for world population growth over short periods of time is given by P P e rt
4-7 Exponential and Logarithmic Equations 33 where P is the population after t years, P is the population at t, and the population is assumed to grow continuously at the annual rate r. How many years, to the nearest year, will it take the world population to double if it grows continuously at an annual rate of %? 84. World Population. Refer to Problem 83. Starting with a world population of 4 billion people and assuming that the population grows continuously at an annual rate of %, how many years, to the nearest year, will it be before there is only 1 square yard of land per person? Earth contains approximately 1.7 1 14 square yards of land. 8. Archaeology Carbon 14 Dating. As long as a plant or animal is alive, carbon 14 is maintained in a constant amount in its tissues. Once dead, however, the plant or animal ceases taking in carbon, and carbon 14 diminishes by radioactive decay according to the equation A A e.14t where A is the amount after t years and A is the amount when t. Estimate the age of a skull uncovered in an archaeological site if 1% of the original amount of carbon 14 is still present. Compute the answer to three significant digits. 86. Archaeology Carbon 14 Dating. Refer to Problem 8. What is the half-life of carbon 14? That is, how long will it take for half of a sample of carbon 14 to decay? Compute the answer to three significant digits. 87. Photography. An electronic flash unit for a camera is activated when a capacitor is discharged through a filament of wire. After the flash is triggered and the capacitor is discharged, the circuit (see the figure) is connected and the battery pack generates a current to recharge the capacitor. The time it takes for the capacitor to recharge is called the recycle time. For a particular flash unit using a 1-volt battery pack, the charge q, in coulombs, on the capacitor t seconds after recharging has started is given by q.9(1 e.t ) How many seconds will it take the capacitor to reach a charge of.7 coulomb? Compute the answer to three significant digits. I R V 88. Advertising. A company is trying to expose a new product to as many people as possible through television advertising in a large metropolitan area with million possible viewers. A model for the number of people N, in millions, who are aware of the product after t days of advertising was found to be N (1 e.37t ) C S How many days, to the nearest day, will the advertising campaign have to last so that 8% of the possible viewers will be aware of the product? 89. Newton s Law of Cooling. This law states that the rate at which an object cools is proportional to the difference in temperature between the object and its surrounding medium. The temperature T of the object t hours later is given by T T m (T T m )e kt where T m is the temperature of the surrounding medium and T is the temperature of the object at t. Suppose a bottle of wine at a room temperature of 7 F is placed in a refrigerator at 4 F to cool before a dinner party. After an hour the temperature of the wine is found to be 61. F. Find the constant k, to two decimal places, and the time, to one decimal place, it will take the wine to cool from 7 to F. 9. Marine Biology. Marine life is dependent upon the microscopic plant life that exists in the photic zone, a zone that goes to a depth where about 1% of the surface light still remains. Light intensity is reduced according to the exponential function I I e kd where I is the intensity d feet below the surface and I is the intensity at the surface. The constant k is called the coefficient of extinction. At Crystal Lake in Wisconsin it was found that half the surface light remained at a depth of 14.3 feet. Find k, and find the depth of the photic zone. Compute answers to three significant digits. 91. Agriculture. Table 1 shows the yield (bushels per acre) and the total production (millions of bushels) for corn in the United States for selected years since 19. Let x represent years since 19. TABLE 1 Yield Total Production Year x (bushels per acre) (million bushels) 19 37.6,78 196 6.6 3,479 197 7 81.4 4,8 198 8 97.7 6,867 199 9 11.6 7,8 Source: U.S. Department of Agriculture. United States Corn Production (A) Find a logarithmic regression model (y a b ln x) for the yield. Estimate (to one decimal place) the yield in 1996 and in 1.
34 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS (B) The actual yield in 1996 was 17.1 bushels per acre. How does this compare with the estimated yield in part A? What effect will this additional 1996 information have on the estimate for 1? Explain. 9. Agriculture. Refer to Table 1. (A) Find a logarithmic regression model (y a b ln x) for the total production. Estimate (to the nearest million) the production in 1996 and in 1. (B) The actual production in 1996 was 7,949 billion bushels. How does this compare with the estimated production in part A? What effect will this 1996 production information have on the estimate for 1? Explain.