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

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  • Thread starter Thread starter Ali Asadullah
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SUMMARY

The equation x^y = y^x has established solutions including (2,4), (4,2), (3,3), and (4.4). Additional solutions exist for the form (3,y) where y can equal 3 or approximately 2.478. The discussion highlights the possibility of complex solutions and emphasizes the need for logarithmic manipulation to explore these relationships further. For a comprehensive understanding, refer to the linked Physics Forums thread.

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Ali Asadullah
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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 [itex]y\approx 2.478[/itex].
And if you take a look at the thread arildno posted, you'll see there are complex solutions as well.
 

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