378 CHAPTER 5 Exponential and Logarithmic Functions In Problems 36–46, solve each equation. Express irrational solutions in exact form. 36. = + 8 4 x 6 3 37. = + 3 3 x x 2 38. = − log 64 3 x 39. = + 5 3 x x 2 40. = − 25 5 x x 2 12 2 41. − = x log 2 2 3 42. = ⋅ 8 4 2 x x5 2 43. ⋅ = 2 5 10 x x 44. ( ) ( ) + + + = x x log 3 log 4 1 6 6 45. = − e 5 x 1 46. + ⋅ − = 9 4 3 3 0 x x 47. Suppose that ( ) ( ) = − + f x x log 2 1. 2 (a) Graph f. (b) What is ( ) f 6 ? What point is on the graph of f ? (c) Solve ( ) = f x 4. What point is on the graph of f ? (d) Based on the graph drawn in part (a), solve ( ) > f x 0. (e) Find ( ) −f x . 1 Graph −f 1 on the same Cartesian plane as f. 48. Amplifying Sound An amplifier’s power output P (in watts) is related to its decibel voltage gain d by the formula = P e 25 d 0.1 (a) Find the power output for a decibel voltage gain of 4 decibels. (b) For a power output of 50 watts, what is the decibel voltage gain? 49. Limiting Magnitude of a Telescope A telescope is limited in its usefulness by the brightness of the star that it is aimed at and by the diameter of its lens. One measure of a star’s brightness is its magnitude; the dimmer the star, the larger its magnitude. A formula for the limiting magnitude L of a telescope—that is, the magnitude of the dimmest star that it can be used to view—is given by = + L d 9 5.1log where d is the diameter (in inches) of the lens. (a) What is the limiting magnitude of a 3.5-inch telescope? (b) What diameter is required to view a star of magnitude 14? 50. Salvage Value The number of years n for a piece of machinery to depreciate to a known salvage value can be found using the formula ( ) = − − n s i d log log log 1 where s is the salvage value of the machinery, i is its initial value, and d is the annual rate of depreciation. (a) How many years will it take for a piece of machinery to decline in value from $90,000 to $10,000 if the annual rate of depreciation is 0.20 (20%)? (b) How many years will it take for a piece of machinery to lose half of its value if the annual rate of depreciation is 15%? 51. Funding a College Education A child’s grandparents purchase a $10,000 bond fund that matures in 18 years to be used for her college education. The bond fund pays 4% interest compounded semiannually. How much will the bond fund be worth at maturity? What is the effective rate of interest? How long will it take the bond to double in value under these terms? 52. Funding a College Education A child’s grandparents wish to purchase a bond that matures in 18 years to be used for her college education. The bond pays 4% interest compounded semiannually. How much should they pay so that the bond will be worth $85,000 at maturity? 53. Estimating the Date When a Prehistoric Man Died The bones of a prehistoric man found in the desert of New Mexico contain approximately 5% of the original amount of carbon-14. If the half-life of carbon-14 is 5730 years, approximately how long ago did the man die? 54. Temperature of a Skillet A skillet is removed from an oven where the temperature is ° 450 F and placed in a room where the temperature is ° 70 F. After 5 minutes, the temperature of the skillet is ° 400 F. How long will it be until its temperature is ° 150 F? 55. World Population The annual growth rate of the world’s population in 2019 was = = k 1.1% 0.011. The population of the world in 2019 was 7,714,576,923. Letting = t 0 represent 2019, use the uninhibited growth model to predict the world’s population in the year 2024. Source: worldometers.info 56. Radioactive Decay The half-life of radioactive cobalt is 5.27 years. If 100 grams of radioactive cobalt is present now, how much will be present in 20 years? In 40 years? 57. Logistic Growth The logistic growth model ( ) = + − P t e 0.8 1 1.67 t 0.16 represents the proportion of new cars with a global positioning system (GPS). Let = t 0 represent 2006, = t 1 represent 2007, and so on. (a) What proportion of new cars in 2006 had a GPS? (b) Determine the maximum proportion of new cars that have a GPS. (c) Using a graphing utility, graph ( ) = P P t . (d) When will 75% of new cars have a GPS? 58. Biology A strain of E-coli Beu 397-recA441 is placed into a nutrient broth at ° 30 Celsius and allowed to grow. The following data are collected.Theory states that the number of bacteria in the petri dish will initially grow according to the law of uninhibited growth.The population is measured using an optical device in which the amount of light that passes through the petri dish is measured. Source: Dr. Polly Lavery, Joliet Junior College Population, y Time (hours), x 0 2.5 3.5 4.5 6 0.09 0.18 0.26 0.35 0.50 (a) Draw a scatter plot treating time as the independent variable.
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