TITLE: Given that $x^y=y^x$, what could $x$ and $y$ be?
QUESTION [13 upvotes]: It's not too difficult to figure out that $x$ and $y$ can both be 1, and also $x$ can be 2 and $y$ can be 4 (and vice versa). But I can't rule out if there are other solutions. Does it have anything to do with inverse functions? Is there a way to see the solutions by graphing, or algebraically?

REPLY [6 votes]: If we convert everything to polar form, we end up with
$$(r\cos\theta)^{r\sin\theta}=(r\sin\theta)^{r\cos\theta}$$
Take the $r$th root of both sides and you'll get
$$(r\cos\theta)^{\sin\theta}=(r\sin\theta)^{\cos\theta}$$
$$r^{\sin\theta}(\cos\theta)^{\sin\theta}=r^{\cos\theta}(\sin\theta)^{\cos\theta}$$
$$r^{\sin\theta-\cos\theta}=\frac{(\sin\theta)^{\cos\theta}}{(\cos\theta)^{\sin\theta}}$$

$$r=\left(\frac{(\sin\theta)^{\cos\theta}}{(\cos\theta)^{\sin\theta}}\right)^{1/(\sin\theta-\cos\theta)}$$

which is the curve you are interested in.  A graph may be found here.
Plugging in $\theta=\arctan(0.5)$ will yield the rectangular coordinate $(4,2)$.