Natural logarithm and pi... help?


by Guero
Tags: logarithm, natural
Guero
Guero is offline
#1
Apr10-05, 09:49 AM
P: 15
I haven't been able to prove:

ln(e)/e > ln(pi)/pi

without calculating any of the values. Help would be much appreciated.
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arildno
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#2
Apr10-05, 09:52 AM
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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.
Guero
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#3
Apr10-05, 09:58 AM
P: 15
mm, I can see that, but I was looking for a proof that shows that e^pi > pi^e

arildno
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#4
Apr10-05, 10:09 AM
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Natural logarithm and pi... help?


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.
Guero
Guero is offline
#5
Apr10-05, 10:13 AM
P: 15
Argh! I get it, Thanks!

I feel pretty stupid now.


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