# 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

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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 $$x = \frac{W(2)}2.$$