Dave Renfro says: >I don't know what Joe's explanation is (when more hand-holding details are included),
Yes, a couple calculus facts are needed, for this proof. I'm haven't learned yet how to "share" a graph with desmos (desmos.com). (No doubt you'ld have to signup to see it.)
In the meantime, I'm sure this description will suffice for the math adept. Draw the line (y=x/e) from 0 through (e,1) and (pi,pi/e). Draw the log curve through the points (e,1) and (pi,log(pi)). The first line is tangent to the log curve at x=e (calculus fact#1). If you've drawn correctly, mentally or on paper or computer, you will see that pi/e > log(pi)*.
pi/e > log(pi)* pi > e log(pi) e^pi > pi^e
*: Analytically, you can use the mean value theorem (calculus fact#2) to *prove* that it is so. Probably can do so just based on the fact that log is continuous with monotonically decreasing derivative, but, eh, life is short.
As I said, there is an even more direct way of seeing this (without using log), but also uses a couple calculus facts.