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Is there any rule for powers so that i can compare which one is greater without actually calculating? For example

54^53 and 53^54 
23^26 and 26^23
3^4 and 4^3 (very simple but how without actually calculating)
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Is it always $a^b$ vs $b^a$? – Aryabhata Dec 30 '10 at 20:18
In my case (GRE preparation), yes it is. – LifeH2O Dec 31 '10 at 18:23
up vote 11 down vote accepted

If $a\gt b\gt e , b^a\gt a^b$. To see this, take logs. You want to compare $a \ln b$ with $b \ln a$. $\ln$ rises slowly, so the larger multiplier wins.

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+1,that's nice explanation,but what could be for any $a^b$ and $c^d$ ? – Quixotic Dec 31 '10 at 10:32
@Debanjan: Comparing logs still makes it easier, but there is no simple answer. Note that given b>d and a, you can find c large enough that c^d>a^b. Sometimes you can still estimate the ratio of logs using whatever you know, like ln 2=.69, or ln 3=1.1 – Ross Millikan Dec 31 '10 at 14:54
if b>d and a>c then a^b>c^d – LifeH2O Dec 31 '10 at 18:32
Problem 99 on the Euler Project site asks you to find the largest of a list of these: – Ross Millikan Jan 1 '11 at 1:08
How one would check that project euler question without calculating? – LifeH2O Jan 1 '11 at 14:11

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