How to simplify $\ln{\left(x + \ln{\left(x + \ln{\left(x + ...\right)}\right)}\right)}$.

Question

I have tried the following:

$s = \ln{\left(x + \ln{\left(x + \ln{\left(x + ...\right)}\right)}\right)}$

$s = \ln{\left(s + x\right)}$

$e^{s} = s + x$

However, I am unsure as to how to proceed.

Thank you.

Answer 1

A solution of $e^s = x+s$ is given by $$ s=-x-W(-e^{-x})\qquad (x\geq 1)$$ where $W$ is the Lambert W-function.