Inverse Exponential Functions
If y = f(x) = ab x , then we may solve for x in terms of y using logarithms:
x = f –1(y) = logb(y/a)
We see that the inverse of an exponential with base b is a logarithm with base b .
Recall that the logarithm is defined only for positive inputs. Thus we must have y/a > 0 for the inverse to exist. Notice, though, that when a > 0 , all outputs y in the forward exponential function f are also positive, and when a < 0 , all outputs y in the forward exponential function f are also negative. In either case, y/a > 0 , and the inverse is defined.
When a and y are positive together, we can use the laws of logarithms to write:
x = f –1(y) = logb(y/a) = logb(y) – logb(a) = logb(y) – k ,
where k = logb(a) .
The only circumstance in which an inverse will not exist is when a = 0 , that is, when the forward exponential degenerates to the constant function y = f(x) = 0 , with a graph that is a horizontal line.