Equations solvable by LambertW

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I am trying to implement equations solvable by LambertW in python, but I am unable to get the desired resources about how to solve such equations. I know what LambertW is, but I want to know:

  • How to determine whether an equation is solvable by LambertW.

  • How to solve an equation by LambertW.

  • Some references that explains its implementation so that it becomes easy for
    me to code.

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I am not sure to be covering all posible cases but, at least, any equation which can write as $$A+Bx+C\log(D+Ex)=0$$ has solutions in terms of Lambert function.

The solution(s) write $$x=-\frac DE+\frac CB\,W(t)\qquad \text{with}\qquad t=\frac B{CE}\,\exp\left(\frac{BD-AE}{CE}\right)$$

Lambert function is implemented in Python