ln(9 4x 5 = ln(75) (4x 5) ln(9) = ln(75) 4x 5 = ln(75) ln(9) ln(75) ln(9) = 1. You don t have to simplify the exact e x + 4e x
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1 Math 11. Exponential and Logarithmic Equations Fall 016 Instructions. Work in groups of 3 to solve the following problems. Turn them in at the end of class for credit. Names. 1. Find the (a) exact solution to the exponential equation 9 4x 5 = 75, and then (b) approximate the solution to 5 decimal places using a calculator. Solution: (a) Using natural logarithms we find ln(9 4x 5 = ln(75) (4x 5) ln(9) = ln(75) 4x 5 = ln(75) ln(9) and so x = 1 ( ) ln(75) 4 ln(9) + 5. (b) Therefore, x = 1 4 ( ) ln(75) ln(9) (a) Use the change of base formula to convert log to a logarithm in base b = 9. (b) Convert log to the natural logarithm, and then approximate it to 5 decimal places. Solution: (a) log = log 9 36 log 9 11 ln 450 (b) log = ln (a) Find the exact solution to the equation ex 0.e x solution. e x + 4e x (b) Use a calculator to express your answer in (a) to six decimal places. = 1. You don t have to simplify the exact 3 Solution: (a) Clearing the denominators and then moving like terms to the same sides yields: 3e x 0.6e x = e x + 4e x 3e x e x = 0.6e x + 4e x (3 1)e x = ( )e x Multiplying both sides by e x / yields (e x )(e x ) = 4.6 e x = 4.6 x = ln ( ) 4.6 x = 1 ( ) 4.6 ln (b) x
2 4. Solve the equation 4 x 5 = 13 3x 4 for x. Leave answer in exact form. Solution: Taking the natural log of both sides of the equation and distributing we obtain (x 5)(ln 4) = ( 3x 4)(ln 13) ( ln 4)x 5 ln 4 = ( 3 ln 13)x 4 ln(13) now bring all the x s to the left side, and all the numbers to the right side to obtain ( ln ln 13)x = 5 ln 4 4 ln 13 and then dividing both sides by ln ln 13 we have x = 5 ln 4 4 ln 13 ln ln A logarithm log a M where M > 0 can always be changed to another base b as follows. Therefore solving this last equation for y yields y = log a M M = a y log b M = y log b a log a M = log b M log b a This is called the change of base formula for logarithms. (a) Use the change of base formula to convert log 7 5 to a logarithm in base b = 11. (b) Convert log 7 00 to the natural logarithm, and then approximate it to 5 decimal places. Solution: (a) log 7 5 = log 11 5 log 11 7 ln 00 (b) log 7 00 = ln Find the exact solution(s) to the equation e x + 80e x = 18. Verify that your solutions work. Solution: Viewing the equation as e x + 80 e x = 18 we multiply both sides of the equation by ex to obtain ( (e x ) e x + 80 ) e x = 18e x (e x ) + 80 = 18e x This is an equation of quadratic form, and the substitution u = e x turns this into a quadratic equation: u + 80 = 18u u 18u + 80 = 0 (u 8)(u 10) = 0 This implies u = 8 or u = 10. Then e x = 8 implies x = ln(8) and e x = 10 implies x = ln(10). So the solutions are x = ln(8), x = ln(10). A check of the solutions is as follows: x = ln(8) e ln(8) + 80e ln(8) = e ln(8) + 80e ln(1/8) = = = 18 Page
3 as desired. as desired. x = ln(10) e ln(10) + 80e ln(10) = e ln(10) + 80e ln(1/10) = = = Solve the equation log (x + 1) + log (x + 5) = log (x + 10) Solution: First, as long as all expressions in the logs are positive, we have log [(x + 1)(x + 5)] = log (x + 10) and so (x + 1)(x + 5) = x + 10 and that implies and then and so potential solutions are x = 5 or x = 1. x + 6x + 5 = x + 10 and so x + 4x 5 = 0 0 = x + 4x 5 = (x + 5)(x 1) = 0 All the logarithms in the original equation are defined when x = 1, so it is a solution; however when x = 5, log(x+1) is not defined since 5+1 < 0. Thus x = 5 is not a solution. Therefore, the only solution is x = Always be careful to check that your solutions work when solving logarithmic equations, because logarithms and their properties are only defined for positive numbers. See what happens when you solve the equation: ln(x) + ln(x 5) = ln(x 10) For this, combine the logs on the left side using: ln(m) + ln(n) = ln(mn) when M > 0 and N > 0, and then use the property ln(e) = ln(f ) implies E = F when ln(e) and ln(f ) are defined. Solution: First, ln(x) + ln(x 5) = ln(x 10) implies ln[x(x 5)] = ln(x 10) and thus x(x 5) = x 10. Then x 5x x + 10 = 0 x 7x + 10 = 0 (x 5)(x ) = 0 Therefore, x = 5 and x = are proposed solutions. However, neither of them works because when x = 5, ln(x 5) = ln(0) which is not defined, and when x =, ln(x 5) = ln( 5) which is undefined since 5 < 0. Thus neither of the proposed solutions work, and so there is no solution. Page 3
4 9. Solve the logarithmic equation + log 6 (3x ) = log 6 (7x + 1). Solution: We can write the equation as ( ) 7x + 1 log 6 (7x + 1) log 6 (3x ) = log 6 3x converting this to exponential form the equation becomes and so 7x + 1 3x = 6 = 7x + 1 3x = 6 7x + 1 = 36(3x ) = 108x 7x 73 = 101x and so x = Therefore, x = 73 is the proposed solution. Because both 3x > 0 and 7x + 1 > 0 when 101 x = 73 73, this means x = is the solution The population of a city is currently and is expected to grow at a rate of 7.5 percent per year for the foreseeable future. Its population is given by P (t) = 35000(1.075 t ) where t is the number of years from today. (a) What will the population be in 9 years? (Express answer as a whole number) (b) At this rate of growth, how long (in years) will it take the population to double? How long (in years) would it take the population to quadruple? Express answers to 1 decimal place. (c) If this growth rate could continue, how long (in years) would it take for the population to reach 3,000,000 people? Express answer to 1 decimal place. Solution: (a) The population will be P (9) = 35000( ) in 9 years. (b) To determine how long it will take the population to double, we solve (35000) = 35000(1.075) t for t. Thus dividing both sides by we have = t and then log() = t log(1.075) t = log() log(1.075) Thus it would take approximately 9.6 years for the population to double. For the population to quadruple, it would have to double twice, so it would take ( ) 19. years for the population to quadruple. (c) To determine how long it will take the population to reach 3,000,000, we solve = 35000(1.075) t for t. Thus dividing both sides by we have /35000 = t and then log( /35000) = t log(1.075) t = log( /35000) log(1.075) Thus it would take approximately 30.7 years for the population to reach 3,000,000 people. Page 4
5 10. The base can always be changed in an exponential function using the property a = b log b a ; and therefore a x = (b log b a ) x = b x log b a. (a) Write the function f(x) = 13 x in as an exponential function in base 9. (b) Write the function g(x) = (13 x ) using the natural base e. Solution: (a) f(x) = 13 x = 9 x log (b) g(x) = e x ln 13. Page 5
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