I haven't been able to prove: ln(e)/e > ln(pi)/pi without calculating any of the values. Help would be much appreciated.
Hint: Consider the function [tex]f(x)=\frac{ln(x)}{x}}[/tex], with domain the positive real half-axis. Determine the function's maximum value.
Well, since you can prove that ln(e)/e is the maximum value for f, we also have: [tex]\pi(ln(e))>eln(\pi)\to{ln}(e^{\pi})>ln(\pi^{e})[/tex] wherefrom your inequality follows.