Symmetry of exponential and logarithmic functions

The exponential function $y = e^x$ and the logarithmic function $y = \ln x$ are inverse functions, and their graphs are symmetric about the line $y = x$.

An inverse function swaps the roles of $x$ and $y$. $y = e^x$ takes $x$ and returns $e^x$; going the other way — recovering $x$ from the value — is $y = \ln x$. For instance $e^x$ passes through $y = 1$ at $x = 0$, and $\ln x$ passes through $y = 0$ at $x = 1$. Points like $(0, 1)$ and $(1, 0)$, with their $x$- and $y$-coordinates swapped, correspond to each other.

In general, the points $(a, b)$ and $(b, a)$ are symmetric about the line $y = x$. Since an inverse swaps $(x, y)$, for every point $(a, b)$ on $y = f(x)$ the inverse's graph contains $(b, a)$. So the two graphs are reflections of each other across $y = x$ as a mirror.

$e^x$ rises steeply with the $x$-axis as its asymptote, while $\ln x$ grows slowly with the $y$-axis as its asymptote, and they are exact mirror images across $y = x$. The large dots on the graph are the symmetric pair $(0, 1)$ and $(1, 0)$.