# Natural logarithm and pi help?

1. Apr 10, 2005

### 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.

2. Apr 10, 2005

### arildno

Hint:
Consider the function
$$f(x)=\frac{ln(x)}{x}}$$,
with domain the positive real half-axis.

Determine the function's maximum value.

Last edited: Apr 10, 2005
3. Apr 10, 2005

### Guero

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

4. Apr 10, 2005

### 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.

5. Apr 10, 2005

### Guero

Argh! I get it, Thanks!

I feel pretty stupid now.

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