Comparing pi to the e with e to the pi

Without a calculator, decide which number is bigger: $\pi^e$ or $e^\pi$. Answer $1$ if $\pi^e$ is bigger and $2$ if $e^\pi$ is bigger.

$2$ ($e^\pi$ is larger). Taking logs and dividing by $e\pi$, compare $\tfrac{\ln e}{e}$ with $\tfrac{\ln\pi}{\pi}$. The function $\tfrac{\ln x}{x}$ peaks at $x=e$, so $\tfrac{\ln e}{e}>\tfrac{\ln \pi}{\pi}$, i.e. $e^\pi>\pi^e$.