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This question already has an answer here:

How can one prove that $e^\pi>\pi^e$ ? Should I use logarithms or something like that? Or is there a gentle smart way of proving so?

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marked as duplicate by Sujaan Kunalan, MJD, MCT, Blue, achille hui Apr 17 '14 at 18:30

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Hint:
$e^n>1+n$ given $n>0$. Now see what happens when $n=\dfrac \pi e-1$.

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  • $\begingroup$ $n$ seems an un-natural choice of variable, no? (+1 though) $\endgroup$ – anaconda Apr 17 '14 at 18:08
  • $\begingroup$ @anaconda Un-natural? Really? :p $\endgroup$ – Hakim Apr 17 '14 at 18:15
  • $\begingroup$ Yeah, I thought it would be a good pun as well as a good point :) $\endgroup$ – anaconda Apr 17 '14 at 18:23

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