Find a value for $\left\{a\mid0

Question

Find a value for $\left\{a\mid0<a<1\right\}$ such that there is no $b\in\mathbb R$ for which $a^a=b^b$ where $a\neq b$.

anonymous · Answer

um... regardless of what a you choose, b =a will work as a choice for b.

anonymous · Answer

Should this problem require that b != a?

anonymous · Answer

Also for an $a\neq b$. Yup lol forgot

anonymous · Answer

I think intuitively I can say that numbers like $\frac{1}{\pi},\frac{1}{e}$ qualify. But I'm not sure how to prove it.

anonymous · Answer

Honestly, this problem might be a little over my head, because it seems like any a would work. Maybe you can give me an example of an a that does not?

anonymous · Answer

$$\left(\frac{1}{2}\right)^\frac{1}{2}=\left(\frac{1}{4}\right)^\frac{1}{4}$$

anonymous · Answer

This is the case for all, I think, non-transcendental numbers since most points are mapped by the function $x^x$.

anonymous · Answer

@FoolForMath Halp me :(