TITLE: Prove that there exists infinitely many pairs of positive real numbers $\alpha$ and $\beta$ such that $\alpha^\alpha = \beta^\beta$
QUESTION [6 upvotes]: I was searching online for some difficult Calculus problems and this one stumped me. 
An example of a pair of numbers would be $\left(\frac{1}{2}, \frac{1}{4}\right)$ because
$$\left( \frac{1}{2} \right)^{\frac{1}{2}} = \left( \frac{1}{4} \right)^{\frac{1}{4}} $$
In the question I was looking at, there was hint given that said to consider the function $f(x) = x^x$ for $x > 0$, and to focus on the interval $(0, 1]$.

REPLY [7 votes]: Consider the function $f(x)=x^x=e^{x\log x}$. Its derivative is
$$
f'(x)=e^{x\log x}\left(\log x+x\frac{1}{x}\right)=x^x(1+\log x)
$$
Can you argue the intervals where $f$ is increasing or decreasing? What can you conclude?

REPLY [2 votes]: A variation on @Lucian's answer:
$$a^a=b^b\quad\Leftrightarrow\quad a\ln a=b\ln b\ .$$
Now plot $f(x)=x\ln x$ and see what you notice.