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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 $b^a$?2010-12-30
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    In my case (GRE preparation), yes it is.2010-12-31

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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 $c^d$ ?2010-12-31
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    @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=1.12010-12-31
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    if b>d and a>c then a^b>c^d2010-12-31
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    Problem 99 on the Euler Project site asks you to find the largest of a list of these: http://projecteuler.net/index.php?section=problems&id=992011-01-01
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    How one would check that project euler question without calculating?2011-01-01
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    That's shit. Now is there any rule to check for any number of base exponent pairs? or one has to make a software to solve the question?2011-01-01
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    Or just paste the columns into your favorite spreadsheet, take the log of the first entry, multiply by the exponent, use copy down to get all the rows, and ask for the maximum. Spreadsheets are very powerful tools.2011-01-01