Properties of Logarithms
1. log_{b}1 = 0 . It follows from the fact that b^{o} = 1.
2. log_{b }b = 1. It follows from the fact that b _{1}= b .
3. log_{b} b^{x} = x . it can be generalized out to b^{log }_{b}^{ f ( x ) }= f ( x ).
4. b ^{logb x} = x . It can be generalized out to b ^{logb f ( x )} = f ( x ) .
Properties 3 and 4 lead to a pleasant relationship among the logarithm & exponential function.
Let's first calculate the following function compositions for f ( x )= b ^{x} and g ( x ) = log_{b} x .
( f o g )( x ) = f [g ( x )] = f (log_{b} x ) = b ^{logb x} = x
( g o f ) ( x ) = g [f ( x )]= g [b ^{x} ] = log _{b} b^{x} = x
Remember again from the section on inverse functions which this means that the exponential & logarithm functions are inverses of each other. It is a nice fact to remember on occasion.
We have to also give the generalized version of Properties 3 & 4 in terms of both the natural and common logarithm
ln e ^{f ( x )} = f ( x) log10 ^{f ( x ) }= f ( x)
e^{ln f ( x ) }= f ( x ) 10^{log f ( x ) }= f ( x )