328 CHAPTER 5 Exponential and Logarithmic Functions The proof uses the fact that = y ax and = y x log a are inverse functions. Proof of Property (1) For inverse functions, ( ) ( ) = − − f f x x x f forall inthedomainof 1 1 Use ( ) = f x ax and ( ) = −f x x log a 1 to find ( ) ( ) = = > − f f x a x x for 0 x 1 loga Now let = x M to obtain = > a M M , where 0. M loga ■ Proof of Property (2) For inverse functions, ( ) ( ) = −f f x x x f forall inthedomainof 1 Use ( ) = f x ax and ( ) = −f x x log a 1 to find ( ) ( ) = = −f f x a x x log for all real numbers a x 1 Now let = x r to obtain = a r log , a r where r is any real number. ■ Using Properties of Logarithms (1) and (2) (a) π = π 2log2 (b) = − − log 0.2 2 0.2 2 (c) = e kt ln kt EXAMPLE 2 Now Work PROBLEM 15 Other useful properties of logarithms are given next. THEOREM Properties of Logarithms In these properties, M , N , and a are positive real numbers, ≠ a 1, and r is any real number. The Log of a Product Equals the Sum of the Logs ( ) = + MN M N log log log a a a (3) The Log of a Quotient Equals the Difference of the Logs ( ) = − M N M N log log log a a a (4) The Log of a Power Equals the Product of the Power and the Log = M r M log log a r a (5) = a e r r a ln (6) We prove properties (3), (5), and (6) and leave the proof of property (4) as an exercise (see Problem 109). Proof of Property (3) Let = A M loga and let = B N log . a These expressions are equivalent to the exponential expressions = = a M a N and A B Now ( ) ( ) = = = + = + + MN a a a A B M N log log log log log a a A B a A B a a Law of Exponents Property (2) of logarithms ■
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