Properties of Logarithms
1. logb1 = 0 . It follows from the fact that bo = 1.
2. logb b = 1. It follows from the fact that b 1= b .
3. logb bx = x . it can be generalized out to blog 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 ) = logb x .
( f o g )( x ) = f [g ( x )] = f (logb x ) = b logb x = x
( g o f ) ( x ) = g [f ( x )]= g [b x ] = log b bx = 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)
eln f ( x ) = f ( x ) 10log f ( x ) = f ( x )