# Fitting a numerical function with another function that depends on parameters

I'd like to fit a branch of the Lambert function :

-ProductLog[-1, x]


in the interval $$[-0.35,0]$$ with a function of the form $$-a\log(-bx)+c+d x$$ or $$-a\log(-x)+c+d x$$ or something like that. I saw that the functions

-Log[-x] + 1.5 + 3 x
-1.5*Log[-x]


were not so bad but I'd like to fit better...

The command $$Fit$$ is not good for that kind of fitting, and Fitting one function to another doesn't look appropriate either.

How could I proceed plz ?

int[a_?NumericQ, c_?NumericQ, d_?NumericQ] :=NIntegrate[( \[Minus]a Log[\[Minus]x] + c + d x -ProductLog[-1, x])^2, {x, -.35, 0}]

Plot[{ \[Minus]a Log[\[Minus]x] + c + d x,ProductLog[-1, x]} /. opt[[2]] // Evaluate, {x, -.35, 0}]