Greg Egan on Nostr: For x>0, define L(x)=log(x)/x. L'(x) = (1-log(x))/x^2 has its sole zero at x=e. So ...

For x>0, define L(x)=log(x)/x.

L'(x) = (1-log(x))/x^2 has its sole zero at x=e.

So L(x) ≤ 1/e, with equality only at x=e.

This means:

e log(x) ≤ x
x^e ≤ e^x

for all x>0, with equality only at x=e.