2010-04-13, 02:47 AM
a^b > b^a
ln(a^b) > ln(b^a) -- assuming both are positive
b ln a > a ln b
(ln a)/a > (ln b)/b
Define f(x) = (ln x)/x
f(a) > f(b)
a^b > b^a if f(a) > f(b).
f(5) = 0.322
f(7) = 0.278
Plot of the function:
![[Image: gif&s=47&w=300&h=173]](http://www1.wolframalpha.com/Calculate/MSP/MSP22419a6ag09a0c5783700004h0bd72egbcdeaff?MSPStoreType=image/gif&s=47&w=300&h=173)
For x > e, f(x) is decreasing, so a^b > b^a if b > a > e.
Anything else in particular?
ln(a^b) > ln(b^a) -- assuming both are positive
b ln a > a ln b
(ln a)/a > (ln b)/b
Define f(x) = (ln x)/x
f(a) > f(b)
a^b > b^a if f(a) > f(b).
f(5) = 0.322
f(7) = 0.278
Plot of the function:
For x > e, f(x) is decreasing, so a^b > b^a if b > a > e.
Anything else in particular?