How do I solve for x? $y=e^x+x+1$

Question

I have no clue really, I know that $x$ can be written as $\ln (e^x)$ and that's it.

Answer 1

Subtract $1$ from both sides & exponentiate each side \begin{eqnarray*} e^{y-1} =e^x e^{e^{x}}. \end{eqnarray*} Now recall the Lambert $W$ function is defined by $we^w=z$ gives $w=W(z)$. So we have \begin{eqnarray*} e^x=W(e^{y-1}) \\ x=\ln(W(e^{y-1})). \\ \end{eqnarray*}