Inverse Logarithmic Functions
If y = f(x) = a logb(x) , then we may solve for x in terms of y using exponentials:
x = f –1(y) = b y/a = (b 1/a) y = k y ,
where k = b 1/a .
We see that the inverse of a logarithm with base b is an exponential with base k = b 1/a .
Recall that both the logarithm and the exponential are defined only for positive bases.
The only circumstance in which an inverse will not exist is when a = 0 and 1/a is undefined; that is, when the forward logarithm degenerates to the constant function y = f(x) = 0 , with a graph that is a horizontal line.