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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 = pi

derivative 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 ) = 0

Well, that means
x - e * ln( x ) > 0, when x not= e

pi - 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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