330 CHAPTER 5 Exponential and Logarithmic Functions CAUTION A common error that some students make is to express the logarithm of a sum as the sum of logarithms. M N M N log is not equal to log log a a a ( ) + + Correct statement MN M N log log log a a a ( ) = + Property (3) Another common error is to express the difference of logarithms as the quotient of logarithms. M N M N log log is not equal to log log a a a a − Correct statement M N M N log log log a a a( ) − = Property (4) CAUTION In using properties (3) through (5), be careful about the values that the variable may assume. For example, the domain of the variable for x loga is x 0 > and for x log 1 a( ) − is x 1. > If these expressions are added, the domain is x 1. > That is, x x x x log log 1 log 1 a a a ( ) ( ) [ ] + − = − is true only for x 1. > j Solution [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + = + − + = + − + + = + − − + = + − − + x x x x x x x x x x x x x x x log 1 1 log 1 log 1 log 1 log log 1 log 1 log log 1 1 2 log 1 3log 4 log 1 a a a a a a a a a a a a 2 3 4 2 3 4 2 3 4 2 1 2 3 4 2 Property (4) Property (3) Property (5) Now Work PROBLEM 51 3 Write a Logarithmic Expression as a Single Logarithm Another use of properties (3) through (5) is to write sums and/or differences of logarithms with the same base as a single logarithm. This skill is needed to solve certain logarithmic equations discussed in the next section. Writing Expressions as a Single Logarithm Write each of the following as a single logarithm. (a) + log 7 4 log 3 a a (b) ( ) − − 2 3 ln8 ln 5 1 2 (c) ( ) + + + − x x log log 9 log 1 log 5 a a a a 2 Solution EXAMPLE 6 (a) ( ) + = + = + = ⋅ = log 7 4 log 3 log 7 log 3 log 7 log 81 log 7 81 log 567 a a a a a a a a 4 (b) ( ) ( ) ( ) ( ) − − = − − = − = = = − = − 2 3 ln8 ln 5 1 ln8 ln 25 1 ln4 ln24 ln 4 24 ln 1 6 ln1 ln6 ln6 2 2 3 (c) ( ) ( ) ( ) [ ] ( ) ( ) + + + − = + + − = + − = ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ x x x x x x x x log log 9 log 1 log 5 log 9 log 1 log 5 log 9 1 log 5 log 9 1 5 a a a a a a a a a a 2 2 2 2 = r M M log log a a r ( ) + = ⋅ M N M N log log log a a a = r M M log log a a r ( ) = = = 8 8 2 4 2 3 3 2 2 ( ) − = M N M N log log log a a a = ln1 0
RkJQdWJsaXNoZXIy NjM5ODQ=