# the lambert function?

I am trying to use the lambert function to solve the following equation for $x$.

$$\frac{a}{b} \ln x-x+ c =0$$

My first step is to rewrite $x$ as $e^{\ln x}$.

$$\frac a b \ln x - e^{ln x} + c =0$$

I then multiply by $b$ and divide by $a$.

$$\ln x - \frac b a e^{\ln x} + \frac b a c =0$$

I have read about the Lambert-W function but am unsure of how to progress!

Write your equation as $$\ln(x) - \frac{bx}{a} = - \frac{bc}{a}$$ take the exponential of both sides: $$x e^{-bx/a} = e^{-bc/a}$$ and multiply by $-b/a$. With $u = -bx/a$ we have $$u e^u = -\frac{b}{a} e^{-bc/a}$$ Thus $u = W\left(-\frac{b}{a} e^{-bc/a}\right)$, and $$x = -\frac{a}{b} W\left(-\frac{b}{a} e^{-bc/a}\right)$$