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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
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closed as off-topic by user296602, Javi, YiFan, Thomas, Shailesh Mar 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 $$x = \frac{W(2)}2.$$

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