SECTION 5.5 Properties of Logarithms 329 Proof of Property (6) Property (1), with = a e, gives = e M M ln Now let = M ar and use property (5). = = e e a a r a r ln ln r ■ Proof of Property (5) Let = A M log . a This expression is equivalent to = a M A Now ( ) = = = = M a a rA r M log log log log a r a A r a rA a Law of Exponents Property (2) of logarithms ■ Now Work PROBLEM 19 2 Write a Logarithmic Expression as a Sum or Difference of Logarithms Logarithms can be used to transform products into sums, quotients into differences, and powers into factors. Such transformations are useful in certain calculus problems. Writing a Logarithmic Expression as a Sum of Logarithms Write ( ) + > x x x log 1 , 0, a 2 as a sum of logarithms. Express all powers as factors. Solution EXAMPLE 3 ( ) ( ) ( ) + = + + = + + = + + x x x x x x x x log 1 log log 1 log log 1 log 1 2 log 1 a a a a a a a 2 2 2 1 2 2 ( ) ⋅ = + M N M N log log log a a a = M r M log log a r a Writing a Logarithmic Expression as a Difference of Logarithms Write ( ) − > x x x ln 1 1 2 3 as a difference of logarithms. Express all powers as factors. Solution EXAMPLE 4 ( ) ( ) ( ) − = − − = − − x x x x x x ln 1 ln ln 1 2 ln 3 ln 1 2 3 2 3 ↑ ↑ ( ) = − = M N M N M r M log log log log log a a a a r a Writing a Logarithmic Expression as a Sum and Difference of Logarithms Write ( ) + + > x x x x log 1 1 0 a 2 3 4 as a sum and difference of logarithms. Express all powers as factors. EXAMPLE 5 (continued)
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