Intermediate Algebra. 8.6 Exponential Equations and Change of Base. Name. Problem Set 8.6 Solutions to Every Odd-Numbered Problem.
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1 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 = ln x = ln Solving the equation: 7. Solving the equation: 5 x = x = 5 ln5 x = ln12 x ln5 = ln12 ln12 x = ln 5 x ln12 = ln 5 x = ln12 ln x = ln 5 ln Solving the equation: 11. Solving the equation: 8 x+1 = 4 2 3x+ 3 = x1 = 4 4 3x + 3 = 2 = 4 1 x 1 = 1 3x = 1 x = 1 x = Solving the equation: 15. Solving the equation: 3 2 x+1 = x = 2 2 x+1 = ln2 ( 2x + 1) = ln2 2x + 1 = ln2 2x = ln2 1 x = 1 ln ' ( x = ln 2 ( 1 2x) = ln 2 1 2x = ln 2 2x = ln 2 1 x = ln ' (
2 17. Solving the equation: 19. Solving the equation: 15 3x 4 = 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 ln x = ln 4 ln 5 x = 1 ln10 3 ln15 + ' ( 1.18 x = ln ln ' ( Solving the equation: 23. Solving the equation: 5 3x2 = x = x = 3 4 4x = 4 x = Solving the equation: 100e ln5 3x2 = ln15 ( 3x 2)ln5 = ln15 3x 2 = ln15 ln5 3x = ln15 ln5 + 2 x = 1 ln15 3 ln5 + ' ( e ln ln
3 27. Solving the equation: ln = = 125 = ln 125 ln = ln Solving the equation: 50e = 32 e = = ln 1 25 ln ln ' ln Evaluating the logarithm: log 8 1 = log1 log Evaluating the logarithm: log 1 8 = log8 log1 = Evaluating the logarithm: log 7 15 = log15 log Evaluating the logarithm: log 15 7 = log 7 log log Evaluating the logarithm: log = log Evaluating the logarithm: log = log 321 log
4 43. Evaluating the logarithm: Evaluating the logarithm: ln Evaluating the logarithm: ln Evaluating the logarithm: ln 45, Using the compound interest formula: 53. Using the compound interest formula: = t = ln = 2 = ln ln = ln2 t ln = ln2 2ln ' ln ' 9.25 It will take years. It will take 9.25 years. 55. Using the compound interest formula: 57. Using the compound interest formula: P = 2P = ln = 2 = ln 2 ln = ln 2 ln 2 4 ln ' 8.75 It will take 8.75 years. ln ln t t = 3 = = 3 = ln = 2 ln ' It was invested years ago.
5 59. Using the continuous interest formula: 1. Using the continuous interest formula: 500e e e ln 2 ln It will take years. 3. Using the continuous interest formula: 1000e e ln2.5 ln It will take years. 5. Using the population model: 32000e e ln 2 ln The city will reach 4,000 toward the end of the year 2007 (October). 7. Using the exponential model: e It will take years. ln1.035 ln t ln1.035 = ln ln 4 ln In the year 2009 it is predicted that 900 million passengers will travel by airline.
6 9. Using the exponential model: ( ) ln1.11 ln t ln1.11 = ln ln ln 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: = = e ln e = ln ln10 ln ln10 = ln ln2 2ln ' It will take years for the money to double. A Coca Cola will cost 1.00 in the year Completing the square: y = 2x 2 + 8x 15 = 2 x 2 + 4x + 4 The lowest point is ( 2, 23). ( ) 8 15 = 2 x Completing the square: y = 12x 4x 2 = 4 x 2 3x ' + 9 = 4 x 3 2 ' The highest point is 3 2, Completing the square: y = 1t 2 = 1 t ( ) + 4 = 1 t 2 ( ) ( ) The object reaches a maximum height after 2 seconds, and the maximum height is 4 feet.
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