Natural logarithm and pi... help?

  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. arildno

    arildno 12,015
    Science Advisor
    Homework Helper
    Gold Member

    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. mm, I can see that, but I was looking for a proof that shows that e^pi > pi^e
     
  5. arildno

    arildno 12,015
    Science Advisor
    Homework Helper
    Gold Member

    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. Argh! I get it, Thanks!

    I feel pretty stupid now.
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook