What are the steps to isolate x in this equation?

I have asked another question with a similar name before. This one is actually difficult.

How do I isolate X in this equation –
y^x – x^y = 0

So far, I have been able to make one side with only x’s, but not a single x. The xth root of x equals the yth root of y.

There seems to almost always be two solutions for x. If y=4, then x can equal 4 or 2. This makes be think that there is a +/- in the answer.

This is NOT homework.