TITLE: How to Compare powers without calculating?
QUESTION [9 upvotes]: 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)

REPLY [11 votes]: 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.