What Are the Solutions to the Equation x^y = y^x?

[Ali Asadullah]

We know that the solutions of x^y=y^x are (2,4) and (4,2), but please tell me how to solve it. I have tried to take log wrt x on both side.

 

[arildno]

Have a look at this thread:
https://www.physicsforums.com/showthread.php?t=412747

 

[Mentallic]

We know that the solutions of x^y=y^x are (2,4) and (4,2)

This is only the case for when restricted to x=2 or y=4. Another example would be (3,3) and (4.4) - of course this could be generalised to all y=x with certain restrictions.

But this isn't the only set of solutions, Taking (3,y) we can have y=3 and also $y\approx 2.478$.
And if you take a look at the thread arildno posted, you'll see there are complex solutions as well.