I am stuck in this question :
If $a+bx =c^x$ then find $x$ in terms of $a, b$ and $c.$
I have tried certain algorithm but it ain't helping also a bit by Lambert function but nothing is happening please help me with a detailed solution any help would be appreciated
You are correct thinking about Lambert function.
What you need to do first is to rewrite the equation as $$a+b x=e^{x \log(c)}$$ and then start the manipulations as described in many examples in the Wikipedia page.
If I am not doing any mistake (please, check all the steps), you whould end with something looking like $$x=-\frac{a}{b}-\frac{W\left(z\right)}{\log (c)}$$ where $z$ is a function of $b$ and $c$.
I let you finishing the work.