Inverse of the Function <em>y = b^x</em>

Function	Type of function	Inverse:	Inverse:
		interchange \(x\) and \(y\)	make \(y\) the subject
\(y = \cfrac{x}{3} + 10\)
\(y = \cfrac{x^{2}}{3}\)
\(y = (10)^{x}\)
\(y = ( \cfrac{1}{3} )^{x}\)

Consider the exponential function

\[y = b^{x}\]

To determine the inverse of the exponential function, we interchange the \(x\)- and \(y\)-variables:

\[x = b^{y}\]

For straight line functions and parabolic functions, we could easily manipulate the inverse to make \(y\) the subject of the formula. For the inverse of an exponential function, however, \(y\) is the index and we do not know a method of solving for the index.

To resolve this problem, mathematicians defined the logarithmic function. The logarithmic function allows us to rewrite the expression \(x = b^{y}\) with \(y\) as the subject of the formula:

\[y = \log_{b}{x}\]

This means that \(x = b^{y}\) is the same as \(y = \log_{b}{x}\) and both are the inverse of the exponential function \(y = b^{x}\).