Is there any "superlogarithm" or something to solve $x^x$? [duplicate]
What you're looking for is the Lambert $W$ function. This is the function such that: $$W(xe^x) = x$$
It does not have a "simple" or explicit form.
To solve your equation, we follow this process: $$x^x = 10$$ $$x\ln x = \ln10$$ $$e^{\ln x}\ln x = \ln10$$ $$W(e^{\ln x}\ln x) = W(\ln10)$$ $$\ln x = W(\ln10)$$ $$x = e^{W(\ln10)}$$
Yes - it's called the Lambert W Function. Scroll down and take a look at Example 2.