Solutions for $x^x=y^y$? How many solutions are there for $x^x=y^y$? I believe they are uncountable, but what is the proof? I have an example stating that $\frac{1}{2}^{\frac{1}{2}}=\frac{1}{4}^{\frac{1}{4}}$. What other examples and proofs of this are there?
The function $x^x$ has a single global minimum at $x = \frac{1}{e}$. You will notice that it is strictly decreasing from $x = 0$ to $x = \frac{1}{e}$ and strictly increasing from $x = \frac{1}{e}$ to infinity.

Moreover it is continuous, so you can use the Intermediate Value Theorem to deduce how many times the function can take on certain values.
The example is one solution.

What about $x = \frac12,$ $y = \frac12$? Is that a solution too? Why or why not?

Can you come up with any other solutions? For how many values of $x$ can you find a solution?