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 $y = x$.

What an inverse function is

An inverse function swaps the roles of $x$ and $y$. The function $y = e^x$ 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.

How to build one

An inverse can be built mechanically. Solving $y = e^x$ for $x$ gives $x = \ln y$; renaming the letters so that the input is again called $x$ gives $y = \ln x$. That recipe, solve and then swap the names, is how inverses are made, and the swap of names is exactly what reflecting the graph across $y = x$ does.

Why the symmetry is about $y = x$

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 has the point $(b, a)$. So the two graphs are reflections of each other in $y = x$ as a mirror.

What gets swapped

Applying the two in succession returns you to the start: $\ln(e^x) = x$ and $e^{\ln x} = x$ say that a round trip changes nothing. That $e^x$ has the $x$-axis as an asymptote while $\ln x$ has the $y$-axis is likewise the same fact seen in the mirror.

Tangent slopes

Tangent slopes correspond as well. The tangent to $e^x$ at $(0, 1)$ has slope $1$, and the tangent to $\ln x$ at $(1, 0)$ also has slope $1$. Reflection turns a slope into its reciprocal, and the reciprocal of $1$ is $1$.

When an inverse exists

Only a one-to-one function has an inverse, and $e^x$ qualifies because it increases monotonically. The symmetry is not special to $e$: $2^x$ and $\log_2 x$ stand in exactly the same relation. The large dots on the graph are the symmetric pair $(0, 1)$ and $(1, 0)$.