TITLE: Is $a^{\ln b} = b^{\ln a}$?
QUESTION [7 upvotes]: I was struggling with a math problem, namely, a limit with a power to the log of something. While I was struggling with it, I found out that $$a^{\ln b} = b^{\ln a}$$ for all positive values that I've tested. Is it true? And if so, can you provide a proof?

REPLY [17 votes]: Do this:
$$a^{\ln(b)} = e^{\ln(a)\ln(b)} = b^{\ln(a)}.$$