Properties of Logarithms

1. log_b1 = 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 )