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HW6 Sol - Math 2312 Homework 6 1 Solve the equation Give an...

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Page 1 of 10 Math 2312 Homework 6 1. Solve the equation. Give an exact answer. 2 ) 8 ( log log 3 3 x x Solution: 1 , 9 0 ) 1 )( 9 ( 0 9 8 9 8 log 3 ) 8 ( ) log( log log 2 ) 8 ( log 2 2 2 3 x x x x x x x x b a c b x x ab b a x x c a where we applied a log property in line one (the sum of logs is a log of a product) and we converted the log equation into an exponential equation in line 2. We also note that x = -1 is not in the domain of x 3 log and ) 8 ( log 3 x in the original equation, so the only solution for the original equation is x = 9. 2. If x x f ln ) ( use properties of logarithms to show that h x h h x f h x f 1 1 ln ) ( ) ( Solution: a h b b a x h x x h x h b a b a h x h x h x h x h x f h x f log log ln ln 1 log log log ln ln ) ln( ) ( ) ( 1 In the second line we applied the property that a difference in logs is the log of a quotient, and in the last line we applied the property that a number times a log is equivalent to placing the number in the exponent of the function inside the log.
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Page 2 of 10 3. Given x x x f ln ) ( and x e x g 2 ) ( , find, and simplify a) )) ( ( x g f Solution: Replacing x in f(x) with empty parentheses, ) ( ln ) ( ) ( f then x e x x f f e e a a e e e e e e e e x g x g x g f x x x bc c b x x x x x x x x 2 1 2 2 1 2 2 2 1 2 2 2 2 2 2 )) ( ( ln ln ln ln ) ( ln ) ( ) ) ( ( ln )) ( ( )) ( ( In the last line, we applied the property x x f f )) ( ( 1 b) )) ( ( x f g Solution: The outer function is now g(x), so we replace all x in g(x) with empty parentheses: ) ( 2 ) ( e g so x x x x a x x x x c b c b x x x x x x x f xe x x f f x e e e b a b e e e e a a a e e e e e x f g 2 1 2 ln 2 ln 2 1 2 2 ln 2 2 ln 2 2 ln 2 2 ) ln ( 2 ) ) ( ( 2 )) ( ( log log ) ) ( ( 2 1
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