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In Reply to: Well... posted by Victor Khomenko on July 25, 2000 at 19:52:00:
Hi, Victor>>>
...my intuitive feel is always to favor the higher power.
<<<Though I agree that e^pi > pi^e, the higher power per se
doesn't guarrantee nothing, 2^3 is LESS than 3^2.
My proof is as follows:e^pi ??? pi^e
ln( e^pi ) ??? ln( pi^e )
pi ??? e * ln( pi )
pi - e * ln( pi ) ??? 0
Now let's consider it as the function
f( x ) = x - e * ln( x ) when x = piderivative is
f'( x ) = 1 - e / x,f( x ) minimum is when f'( x ) = 0, when x = e,
min value is
f( e ) = e - e * ln( e ) = 0Well, that means
x - e * ln( x ) > 0, when x not= epi - e * ln( pi ) > 0
pi > e * ln( pi )
ln( e^pi ) > ln( pi^e )
e^pi > pi^e
Hope my deduction is correct. :)
regards, gnat
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Follow Ups
- not so well ;-) - gnat 00:24:12 07/26/00 (7)
- Got a closet big enough? - Victor Khomenko 05:57:57 07/26/00 (6)
- I'll use my dacha :) - gnat 00:37:21 07/27/00 (5)
- Your papa member of tse-ka? - Victor Khomenko 06:23:52 07/27/00 (4)
- nope, he's from tsa-gi :) - gnat 23:51:29 07/27/00 (3)
- I don't think this is the soln. you're looking for but it works... - Shane 13:46:28 07/28/00 (1)
- minor correction - Shane 13:51:09 07/28/00 (0)
- Right kind of dad - Victor Khomenko 06:39:12 07/28/00 (0)
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