# Solving an equations using LambertW function [closed]

I have just started learning the LambertW function, so if my question is very basic I am really sorry but I can't understand how solving for x in the below equation

ln(x) + 2*x = 0


gave

LambertW(2)/2


## closed as off-topic by user296602, Javi, YiFan, Thomas, ShaileshMar 28 at 0:06

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Community, Javi, YiFan, Thomas, Shailesh
If this question can be reworded to fit the rules in the help center, please edit the question.

## 1 Answer

The Lambert-W function is the inverse of the function $$f(w) = we^w$$, $$w \ge -1$$. That is to say, $$W(z)e^{W(z)} = z$$ for $$z \ge - \dfrac 1e$$.

Say you wish to solve $$\ln x + 2x = 0$$. To bring the function $$we^w$$ into the picture, exponentiate the original expression to get $$x e^{2x} = e^{\ln x + 2x} = e^0 = 1$$ so that $$2x e^{2x} = 2.$$ With $$z = 2$$ you obtain $$W(2) = 2x$$ so that $$x = \frac{W(2)}2.$$