How to find what number raised to its own power is equal to a given number?

Question

If $x^x = 3$, how can I find x? I know I can rearrange this to $\log_x(3) = x$, and that some calculators can solve this, but how would you do this manually?

Answer 1

First, we have $x \ln x = \ln 3$. Here, $\ln x$ represents the natural logarithm. So $$e^{\ln }\ln = \ln 3.$$ Therefore, \begin{align} \ln x &= W( \ln 3 ) \\\implies x &= e^{W( \ln 3)} \end{align}