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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?


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

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

  • $\begingroup$ $n$ 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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