Is $a^{\ln b} = b^{\ln a}$?

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?

• Take log on both sides and compare. – Aryabhata Dec 18 '15 at 2:20
• Thanks! I guess it was pretty obvious in hindsight. ;-) – Drew Christensen Dec 20 '15 at 17:48

$$a^{\ln(b)} = e^{\ln(a)\ln(b)} = b^{\ln(a)}.$$