# Power trip

Here’s a mathematical paper of which I can smugly say I understood every word.

Ha ha. No, seriously, I like it. A visual proof that $e^\pi>\pi^e$.

But as Math with Bad Drawings pointed out someone else pointed out, the proof doesn’t depend on $\pi$ but can be generalized to any number $> e$, and as Math with Bad Drawings pointed out, a similar proof works for any positive number $< e$. That is, for all $a > 0$ and $a\ne e$, $e^a>a^e$.

Proof here if you don’t want to click through to MwBD:

If $a > e$,

$\displaystyle \frac{1}{e}(a-e) = \frac{a}{e}-1 > \int_e^a \frac{1}{x}dx = \ln(a)-\ln(e) = \ln(a)-1$

so

$\displaystyle \frac{a}{e} > \ln(a)$

or

$\displaystyle a > \ln(a^e)$

or

$\displaystyle e^a > a^e$

and similarly, if $b < e$,

$\displaystyle \frac{1}{e}(e-b) = 1-\frac{b}{e} < \int_b^e \frac{1}{x}dx = \ln(e)-\ln(b) = 1-\ln(b)$

so

$\displaystyle \frac{b}{e} > \ln(b)$

or

$\displaystyle b > \ln(b^e)$

or

$\displaystyle e^b > b^e$

QED