Exponential equation with a trap that seems impossible to escape

Question

How would you go about solving this equation: $x^x=10$.

The much I've figured is $x\ln{x}=\ln{10}$

Answer 1

Take that last equation you have and notice that

$$\log 10 = x \log x = e^{\log x} \log x$$

At this point we can apply the inverse of $ze^z$ to both sides, the Lambert W function.

$$W(\log 10) = W\left(\log x e^{\log x}\right) = \log x$$

which means $x = e^{W(\log 10)}$

Answer 2

We get $x=\frac{\ln10}{\ln x} $ Now use numerical methods to approximate the answer.