Nonhomogeneous equation involving logarithm

Question

This is probably a lame question, but what is the general approach to solving $\log z +z \sigma +1=0$ for $z$? Wolfram Alpha obtains a Lambert W-function, but I don't quite see how.

Answer 1

Assuming $\sigma\neq 0$, you need something like Lambert's W function.

Taking exponentials of both sides, we have $$\begin{align*} e^{\log z + z\sigma + 1} &= 1\\ e^{\log z}e^{\sigma z} e&= 1\\ ze^{\sigma z}&= \frac{1}{e}\\ \sigma ze^{\sigma z} &= \frac{\sigma}{e} \end{align*}$$ Now let $x=\sigma z$. Then the equation is equivalent to $$xe^x = \frac{\sigma}{e},$$ which means that $x=W(\frac{\sigma}{e})$, hence $$z = \frac{x}{\sigma} = \frac{1}{\sigma} W\left(\frac{\sigma}{e}\right).$$

(If $\sigma=0$, then you just have $\log z + 1 = 0$, or $z=e^{-1}$.)