3-4 Exponential and Logarithmic Equations
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1 Solve each equation x + 1 = 3 x x + 2 = 2 5x x + 6 = 4 2x ASTRONOMY The brightness of two celestial bodies as seen from Earth can be compared by determining the variation in brightness between the two bodies. The variation in brightness V can be calculated by the brighter body. where m f is the magnitude of brightness of the fainter body and m b is the magnitude of brightness of esolutions Manual - Powered by Cognero Page 1
2 a. The Sun has m = 26.73, and the full Moon has m = Determine the variation in brightness between the Sun and the full Moon. b. The variation in brightness between Mercury and Venus is Venus has a magnitude of brightness of 3.7. Determine the magnitude of brightness of Mercury. c. Neptune has a magnitude of brightness of 7.7, and the variation in brightness of Neptune and Jupiter is 15,856. What is the magnitude of brightness of Jupiter? a. b. c. esolutions Manual - Powered by Cognero Page 2
3 Solve each equation. 51. e 2x 15e x + 56 = e 2x 5e x = 6 The logarithm of a negative number provides no real solution, so x esolutions Manual - Powered by Cognero Page 3
4 55. 8e 4x 15e 2x + 7 = 0 Solve each logarithmic equation. 61. ln x + ln (x + 7) = ln 18 Substitute into original equation to eliminate extraneous solutions. The logarithm of a negative number provides no real solution, so x cannot equal 9. The solution is 2. esolutions Manual - Powered by Cognero Page 4
5 63. ln (x 3) + ln (2x + 3) = ln ( 4x 2 ) Substitute into original equation to eliminate extraneous solutions. The logarithm of a negative number provides no real solution, so there is no solution. 65. log (x + 6) = log (8x) log (3x + 2) Substitute into original equation to eliminate extraneous solutions. The logarithm of a negative number provides no real solution, so there is no solution. esolutions Manual - Powered by Cognero Page 5
6 67. ln (3x 2 4) + ln (x 2 + 1) = ln (2 x 2 ) Solve each logarithmic equation. 69. log 2 (2x 6) = 3 + log 2 x Substitute into original equation to eliminate extraneous solutions. The logarithm of a negative number provides no real solution, so there is no solution. esolutions Manual - Powered by Cognero Page 6
7 71. log x = 1 log (x 3) Substitute into original equation to eliminate extraneous solutions. 2 3 = 5 The logarithm of a negative number provides no real solution, so x = log 9 9x 2 = log 9 x Solve each logarithmic equation. 75. log (29,995x + 40,225) = 4 + log (3x + 4) esolutions Manual - Powered by Cognero Page 7
8 80. TECHNOLOGY A chain of retail computer stores opened 2 stores in its first year of operation. After 8 years of operation, the chain consisted of 206 stores. a. Write a continuous exponential equation to model the number of stores N as a function of year of operation t. Round k to the nearest hundredth. b. Use the model you found in part a to predict the number of stores in the 12 th year of operation. a. The general continuous exponential equation is N = N 0 e kt. The initial number of stores N 0 is 2. At year 1, t = 1 and N = N 0 = 2, so the equation modeling the situation is. Use the data to solve for k. The equation is. b. esolutions Manual - Powered by Cognero Page 8
9 81. STOCK The price per share of a coffee chain s stock was $0.93 in a month during its first year of trading. During its fifth year of trading, the price per share of stock was $3.52 during the same month. a. Write a continuous exponential equation to model the price of stock P as a function of year of trading t. Round k to the nearest ten-thousandth. b. Use the model you found in part a to predict the price of the stock during the ninth year of trading. a. The general continuous exponential equation is P = P 0 e kt. The initial stock price P 0 is $0.93 or At year 1, t = 1 and P = P 0 = 0.93, so the equation modeling the situation is. Use the information to solve for k. The equation is P = 0.93e (t 1). b. Solve each logarithmic equation. 83. esolutions Manual - Powered by Cognero Page 9
10 log 3 x = 60 esolutions Manual - Powered by Cognero Page 10
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