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Prove that: e^pi > pi^e, without using a calculator

  1. Sep 24, 2004 #1
    How can you prove that:
    e^pi > pi^e,
    without using a calculator. (all you know is the values for e and pi.)
     
  2. jcsd
  3. Sep 24, 2004 #2

    mathwonk

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    well that says that e^pi > e^(e ln(pi)), so since e^x is increasing you just have to show that pi > e ln(pi). so we just need to show that pi/ln(pi) > e.

    but x/ln(x) is increasing for x >=e, and e/ln(e) = e, and pi > e, so pi/ln(pi) > e/ln(e) = e. qed.

    thats cute, i'm gonna try that on my calculus class.
     
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