7
$\begingroup$

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?

$\endgroup$
  • 10
    $\begingroup$ Take log on both sides and compare. $\endgroup$ – Aryabhata Dec 18 '15 at 2:20
  • $\begingroup$ Thanks! I guess it was pretty obvious in hindsight. ;-) $\endgroup$ – Drew Christensen Dec 20 '15 at 17:48
17
$\begingroup$

Do this:

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.