I noticed the following question generated quite a lot of interest on the Internet. Which is larger ##e^{\pi}## or ##\pi^e##?
Most (if not all) solutions focus on this as a problem for these specific numbers. And, in fact, the same question was then asked for other specific pairs of numbers. Moreover, most of the solutions seemed to be quite complicated. So, I thought I would post my solution here.
First, since the natural log is an increasing function, we have for any ##a, b > 0##...