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Unformatted text preview: Logarithm and Exponential Functions We want to give a precise definition for the logarithm and derive its properties. The exponential function is the inverse function for the logarithm. Based on properties of the logarithm, the properties of the exponential function then follow. Logarithm. We define the logarithm by the following formula. (1) ln( x ) = 1 t dt 1 x ! for all x > The following two properties follow immediately. (2) d dx ln( x ) = 1 x (3) ln(1) = We now show the following property. (4) ln( x ! y ) = ln( x ) + ln( y ) Let a > be any constant. Consider the function ln( a ! x ) . If we compute the derivative we get d dx ln( a ! x ) = 1 x . Since this is also the derivative of ln( x ) , these two functions must differ by a constant. So, ln( A ! x ) " ln( x ) + C . We can determine the value of C by letting x = 1 . With that substitution we get ln( A ) = ln(1) + C = + C = C . Consequently, ln( a ! x ) = ln( a ) + ln( x ) . This proves formula (4)....
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