$x^x=8$
We must find the only one irrational solution, and prove that this solution is the only one. The uniqueness of the root seems obvious to me from the monotonically increasing graph of the $x^x$, but I don't know how to solve. I've tried messing around with logarithms, but it didn't work. So far my best guess was to note that
$2,3884^{2,3884}=7,99964856151$
and try to find an expression involving roots of naturals that would be close to $2,3884$, but there are way too many numbers.
Thank you.
Irrationality of the root: If it was rational, say $\frac{a}{b}$ (both $a$ and $b$ positive natural numbers), we would have
$$\left( \frac{a}{b} \right ) ^{\frac{a}{b}} = 8$$
$$\left( \frac{a}{b} \right )^a = 8^b$$
But this would mean that a positive natural power of the rational number $\frac{a}{b}$ is a natural number, so it must be a natural number too, so $b=1$. But this leads to the contradiction
$$a^a = 8$$
Only thing $a$ can be is $1, 2, 4, 8$ (because $8$ doesn't have any other positive divisors) but none of these work.