Introduction to Exponential and Logarithmic Functions

The inverse of an exponential function is a logarithmic function and vice versa. That is, the two functions undo each other. Thus $log_{b}b^{x}=x $ and $b^{log_{b}x}=x $. Composing the functions in either order leaves the initial input unchanged. 

Another way of saying this is that if $f(x)=log_{b}x $, then the inverse function is given by $f^{-1}(x)=b^{x} $, and vice versa.

Lastly, as with all inverse functions, if we graph $f(x)=log_{b}x $  and $f^{-1}(x)=b^{x}$ on the same plane, the graphs will be symmetric across the line $y=x$. That is, if we fold the plane over the line $y=x$, the two curves will lie on each other. Another way of thinking about this is that if we generate points on the curve of $f(x)=log_{b}x$ we can find the points on the curve of  $f^{-1}(x)=b^{x}$ by interchanging the $x$ and $y$ coordinates of the points.

In the following graph you can see an exponential function in red and its inverse, a logarithmic function, in blue. The graphs are symmetric over the line $y=x$, which is pictured in black. Further, a point $(t,u=b^t)$ on the graph of $f(x)$ yields a point $(u,t=log{_b}u)$ on the graph of the logarithm and vice versa. 

Logarithm function

The graph of the logarithm function $log_b(x)$ (blue) is obtained by reflecting the graph of the function $b(x)$ (red) at the diagonal line ($x=y$).

Thus far we have been looking at logs of the base $b$. Let us consider instead the natural log (a logarithm of the base $e$).  The natural logarithm is the inverse of the exponential function $f(x)=e^x$. It is defined for $e>0$, and satisfies $f^{-1}(x)=lnx $.

As they are inverses composing these two functions in either order yields the original input. That is, $e^{lnx}=lne^x=x$.