In 1768, Johann Lambert showed that if $x$ is a nonzero rational number, then neither $e^{x}$ nor $\tan x$ can be rational. And so, for example, because $\tan \frac{\pi}{4} = 1$, we may infer that $\frac{\pi}{4}$, and in particular, $\pi$, is irrational.
My question is, are there other functions $f(x)$ similar to $e^{x}$ and $\tan x$ from which we may infer that if $x$ is nonzero and rational then $f(x)$ is irrational?