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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)
LifeH2O
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1 Answers1

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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.

Ross Millikan
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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: http://projecteuler.net/index.php?section=problems&id=99 – Ross Millikan Jan 01 '11 at 01:08
  • How one would check that project euler question without calculating? – LifeH2O Jan 01 '11 at 14:11
  • 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? – LifeH2O Jan 01 '11 at 14:17
  • 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. – Ross Millikan Jan 01 '11 at 16:36