Solving for $t$ in $t=x^t$

Question

Is it possible to solve for $t$ in $t=x^t$? Using log on both sides does not seem to help. $$\log t=t\log x$$ $$\log x=\frac{\log t}t$$

Answer 1

This is done using Lambert's W function. The solution for this particular equation is

$$t = \frac{W\left(-\log x\right)}{-\log x} = e^{-W(-\log x)}=h(x),$$

where $h$ is Euler's iterated exponential.