Can You Prove ln(e)/e > ln(pi)/pi Without Calculations?

[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
$$f(x)=\frac{ln(x)}{x}}$$,
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:
$$\pi(ln(e))>eln(\pi)\to{ln}(e^{\pi})>ln(\pi^{e})$$
wherefrom your inequality follows.

 

[Guero]

Argh! I get it, Thanks!

I feel pretty stupid now.