8. Exponential Equations and Change of Base 1. Solving the equation: 3. Solving the equation: 3 x = 5 5 x = 3 x = ln5 x = ln5 ln5 x = x ln5 = x = ln5 1.450 x = ln5 0.82 5. Solving the equation: 7. Solving the equation: 5 x = 12 12 x = 5 ln5 x = ln12 x ln5 = ln12 ln12 x = ln 5 x ln12 = ln 5 x = ln12 ln5 1.5440 x = ln 5 ln12 0.477 9. Solving the equation: 11. Solving the equation: 8 x+1 = 4 2 3x+ 3 = 2 2 4 x1 = 4 4 3x + 3 = 2 = 4 1 x 1 = 1 3x = 1 x = 1 x = 2 3 13. Solving the equation: 15. Solving the equation: 3 2 x+1 = 2 3 12 x = 2 2 x+1 = ln2 ( 2x + 1) = ln2 2x + 1 = ln2 2x = ln2 1 x = 1 ln 2 2 1 ' ( 0.1845 12 x = ln 2 ( 1 2x) = ln 2 1 2x = ln 2 2x = ln 2 1 x = 1 2 1 ln ' ( 0.1845
17. Solving the equation: 19. Solving the equation: 15 3x 4 = 10 52 x = 4 ln15 3x 4 = ln10 ( 3x 4)ln15 = ln10 ln 52 x = ln 4 ( 5 2x)ln = ln 4 3x 4 = ln10 5 2x = ln 4 ln15 ln 3x = ln10 ln15 + 4 2x = ln 4 ln 5 x = 1 ln10 3 ln15 + ' ( 1.18 x = 1 2 5 ln ln ' ( 2.1131 21. Solving the equation: 23. Solving the equation: 5 3x2 = 15 3 4 x = 81 3 4 x = 3 4 4x = 4 x = 1 25. Solving the equation: 100e 3 250 ln5 3x2 = ln15 ( 3x 2)ln5 = ln15 3x 2 = ln15 ln5 3x = ln15 ln5 + 2 x = 1 ln15 3 ln5 + ' ( 1.2275 e 3 5 2 3 ln 5 2 1 3 ln 5 2 0.3054
27. Solving the equation: 1200 1 + 0.072 1 + 0.072 4 ln 1 + 0.072 = 25000 = 125 = ln 125 ln 1 + 0.072 = ln 125 29. Solving the equation: 50e 0.074 = 32 e 0.074 = 1 25 0.074 = ln 1 25 ln 125 4 ln 1 + 0.072 ' 42.5528 ln 1 25 0.0742.0147 31. Evaluating the logarithm: log 8 1 = log1 log8 1.3333 33. Evaluating the logarithm: log 1 8 = log8 log1 = 0.7500 35. Evaluating the logarithm: log 7 15 = log15 log 7 1.3917 37. Evaluating the logarithm: log 15 7 = log 7 log15 0.718 log 240 39. Evaluating the logarithm: log 8 240 = log 8 2.35 41. Evaluating the logarithm: log 4 321 = log 321 log 4 4.132
43. Evaluating the logarithm: 45 5.8435 45. Evaluating the logarithm: ln 0.345 1.042 47. Evaluating the logarithm: ln10 2.302 49. Evaluating the logarithm: ln 45,000 10.7144 51. Using the compound interest formula: 53. Using the compound interest formula: 500 1+ 0.0 = 1000 1000 1+ 0.12 t = 3000 1+ 0.0 2 ln 1+ 0.0 = 2 = ln2 1+ 0.12 ln 1+ 0.0 = ln2 t ln 1+ 0.12 = ln2 2ln 1+ 0.0 ' 11.72 ln 1+ 0.12 ' 9.25 It will take 11.72 years. It will take 9.25 years. 55. Using the compound interest formula: 57. Using the compound interest formula: P 1 + 0.08 = 2P 25 1 + 0.0 = 75 1 + 0.08 4 ln 1 + 0.08 = 2 = ln 2 ln 1 + 0.08 = ln 2 ln 2 4 ln 1+ 0.08 ' 8.75 It will take 8.75 years. ln 1+ 0.12 1 + 0.0 2 ln 1 + 0.0 t t = 3 = = 3 = ln 1 + 0.0 = 2 ln 1 + 0.0 ' 18.58 It was invested 18.58 years ago.
59. Using the continuous interest formula: 1. Using the continuous interest formula: 500e 0.0 1000 500e 0.0 1500 e 0.0 2 0.0 ln 2 ln2 0.0 11.55 It will take 11.55 years. 3. Using the continuous interest formula: 1000e 0.08 2500 e 0.08 2.5 0.08 ln2.5 ln2.5 0.08 11.45 It will take 11.45 years. 5. Using the population model: 32000e 0.05 4000 e 0.05 2 0.05 ln 2 ln 2 0.05 13.9 The city will reach 4,000 toward the end of the year 2007 (October). 7. Using the exponential model: 4 1.035 900 1.035 900 4 e 0.0 3 0.0 0.0 18.31 It will take 18.31 years. ln1.035 ln 900 4 t ln1.035 = ln 900 4 900 ln 4 ln1.035 19 In the year 2009 it is predicted that 900 million passengers will travel by airline.
9. Using the exponential model: 78.1 1.11 ( ) 800 1.11 800 78.1 ln1.11 ln 800 78.1 t ln1.11 = ln 800 78.1 ln 800 78.1 ln1.11 22 In the year 1992 it was estimated that 800 billion will be spent on health care expenditures. 71. Using the compound interest formula: 73. Using the exponential formula: 1552 1+ 0.07 = 33104 1+ 0.07 = 2 0.10e 0.057 1.00 ln 1+ 0.07 e 0.057 10 = ln2 0.057 ln10 ln 1+ 0.07 ln10 = ln2 0.057 40 ln2 2ln 1+ 0.07 ' 10.07 It will take 10.07 years for the money to double. A Coca Cola will cost 1.00 in the year 2000. 75. Completing the square: y = 2x 2 + 8x 15 = 2 x 2 + 4x + 4 The lowest point is ( 2, 23). ( ) 8 15 = 2 x + 2 77. Completing the square: y = 12x 4x 2 = 4 x 2 3x + 9 4 ' + 9 = 4 x 3 2 ' The highest point is 3 2,9. 79. Completing the square: y = 1t 2 = 1 t 2 + 4 ( ) + 4 = 1 t 2 ( ) 2 23. 2 + 9. ( ) 2 + 4 The object reaches a maximum height after 2 seconds, and the maximum height is 4 feet.