How do I prove that, for example $$e^\pi \pi^e$$

Question

How do I prove that, for example $$e^\pi > \pi^e$$

fibonaccichick666 · Answer

if you take the natural log of both sides you can get something that should be easier to solve

anonymous · Answer

Well it's not that I want to "solve it", since it's just real numbers. I want to give a proof for why the above statement is true. 

And to secure the integrity of the proof, of course I cannot use what I want to proof as an assumption, as that would simply lead me to believe it is true in any case, even if it should be wrong.

fibonaccichick666 · Answer

ahh ok then I'd generalize it do $e^n>n^e$ and test for n=e, n>e, and n<e

fibonaccichick666 · Answer

then compare graphs maybe, or start with something that is trivially true and flow into this

fibonaccichick666 · Answer

Not sure, for my classes I was allowed a direct pf on this one

anonymous · Answer

I'd love to start off with something trivially true and "flow" into this. In such a case, though, my problem is that I don't know what that trival part would be.

fibonaccichick666 · Answer

idk I'll think on it gtg to class

john_es · Answer

You can try a serial expansion (McLaurin series) of both sides as if they were functions, until first order. Then compare.

john_es · Answer

Better look here, http://answers.yahoo.com/question/index?qid=20110103144014AAaEFMk