Natural logarithm and pi... help?

Guero

I haven't been able to prove:

ln(e)/e > ln(pi)/pi

without calculating any of the values. Help would be much appreciated.

arildno

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.

Guero

mm, I can see that, but I was looking for a proof that shows that e^pi > pi^e

arildno

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.

Guero

Argh! I get it, Thanks!

I feel pretty stupid now.