How to approximate
$y=\frac{W(e^{cx+d})}{W(e^{ax+b})}$
with (a) simple function(s)?
given $a=-1/\lambda_0$, $b=(\mu_0+\lambda_0)/\lambda_0$, $c=1/\lambda_1$, $d=(\mu_1+\lambda_1-1)/\lambda_1$ for positive $\mu_0,\lambda_0,\mu_1,\lambda_1$
where $W$ is a Lambert $W$ function, i.e., if $y=xe^x$ then $x=W(y)$
My problem is that I can not invert the function and get $x=f(y)$ alone and decided to go for some nice approximations.
Thanks alot for any help.