I am trying to solve the following equation, where $0 \leq a < b \leq 1$ are constants and $x \in (a,b)$; $\frac{x-a}{b-a} = e^{-2\log(2)/(x+1)}$ and stumbled across the Lambert W-function which I can use if I can transform my equation into something of the form $m = z^z$ where $m$ is some constant and $x$ some functions which can be expressed in terms of x. Is this possible?
I can get this far: $\frac{x-a}{b-a} = e^{-2\log(2)/(x+1)} \Leftrightarrow [x+1 = w] \Leftrightarrow$ $\frac{w-a-1}{b-a} = e^{-2\log(2)/w} \Leftrightarrow [A = (b-a)e^{-2\log(2)}, B=-(a+1)] \Leftrightarrow $ $w+B = Ae^{1/w}$ but do not know how to continue from this.