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

compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ?
Another proof uses the fact that $\displaystyle \pi \ne e$ and that $e^x &gt; 1 + x$ for $x \ne 0$.
We have $$e^{\pi/e -1} &gt; \pi/e,$$
and so
$$e^{\pi/e} &gt; \pi.$$
Thus,
$$e^{\pi} &gt; \pi^e.$$
Note: This proof is not specific to $\pi$.
$$e^\pi&gt;\pi^e$$ because if we subtract $e$ from both exponents we get $$e^{\pi-e}&gt;1$$ which is true because $e$ is is greater than $1$ so when it 8s raised to that power it is equal to $$\frac{e^\pi}{e^e}$$ and that has to be greater than 1 because the top is greater than the bottom.