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**Unformatted text preview: **Logarithmic and Exponential Equations and Inequalities Theorem: If N = M, then N M a a log log = Theorem: If , then N = M. N M a a log log = These properties become critical when solving exponential and logarithmic equations and inequalities. From the graphs of the logarithmic functions seen before, we know that they are one-to-one functions. Hence we have the next two properties. Useful Results Solving Exponential Equations places. decimal three answer to your Round . 15 2 equation the Solve 1 3 = + x 15 is NOT a power of 2, so we cant solve the equation using the properties of exponents. Using logarithms we can solve equations like the one given. We can use any base, but base 10 or e are built in the calculators, so use one of them. 15 2 1 3 = + x ( 29 ) 15 ln( 2 ln 1 3 = + x Using the properties of logarithms: ) 15 ln( ) 2 ln( ) 1 3 ( = + x 15 ln 2 ln ) 2 ln 3 ( = + x 2 ln 15 ln ) 2 ln 3 (- = x 969 . 2 ln 3 2 ln 15 ln - = x Example Solve: 7 5 3 2 x x =- ln ln 7 5 3 2 x x =- ( 29 x x ln ln 7 3 2 5 =- ( 29 x x ln ln ln 7 3 5 2 5 =- ( 29 x ln ln ln 7 3 5 2 5- = - 117 . 1 5 ln 3 7 ln 5 ln 2 -- = x Example Solve:...

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