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Natural logarithm and pi... help? 
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#1
Apr1005, 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. 


#2
Apr1005, 09:52 AM

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Hint:
Consider the function [tex]f(x)=\frac{ln(x)}{x}}[/tex], with domain the positive real halfaxis. Determine the function's maximum value. 


#3
Apr1005, 09:58 AM

P: 15

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



#4
Apr1005, 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. 


#5
Apr1005, 10:13 AM

P: 15

Argh! I get it, Thanks!
I feel pretty stupid now. 


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