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Natural logarithm and pi help?

  1. Apr 10, 2005 #1
    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. jcsd
  3. Apr 10, 2005 #2

    arildno

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

    Determine the function's maximum value.
     
    Last edited: Apr 10, 2005
  4. Apr 10, 2005 #3
    mm, I can see that, but I was looking for a proof that shows that e^pi > pi^e
     
  5. Apr 10, 2005 #4

    arildno

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    Well, since you can prove that ln(e)/e is the maximum value for f, we also have:
    [tex]\pi(ln(e))>eln(\pi)\to{ln}(e^{\pi})>ln(\pi^{e})[/tex]
    wherefrom your inequality follows.
     
  6. Apr 10, 2005 #5
    Argh! I get it, Thanks!

    I feel pretty stupid now.
     
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