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

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The equation x^y = y^x has known solutions including (2,4), (4,2), and (3,3), with additional solutions like (3, y) where y can be approximately 2.478. The discussion highlights that solutions can be generalized to y=x under certain restrictions. Complex solutions also exist, as referenced in an external thread. Overall, the equation presents a variety of solutions beyond the commonly cited pairs.
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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.
 
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Ali Asadullah said:
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.
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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