Comparing $\pi^e$ and $e^\pi$ without calculating them How can I

compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ?


Let
$$f(x) = e^x$$
$$G(x) = x^e$$
We can simply show that
$$f(e)=G(e)$$
$$f'(e)=G'(e)$$
For $x > e$ the $f(x)$ will grow faster than $G(x)$
Then
$$e^{\pi} > \pi^{e}$$
