I have been trying to find a solution for equations of the type $x^{px-c} = a$. I know how to use Lambert W. function to find solutions for $x^x = a$, but the function of $x$ at the exponent is making things much difficult for me. I do not how to proceed in this scenario. Should I try series expansion or some approximation method?
Any help will be greatly appreciated.
I do not think that explicit solutions could be obtained for the zero's of $$f(x)=x^{p\,x-c} - a$$ and you will, more than likely, use numerical methods.
In the case where $c=0$, there is a solution, still expressed in terms of Lambert function. It would be $$x=\frac{\log (a)}{p\, W\left(\frac{\log (a)}{p}\right)}$$ obtained writing $x^x=a^{\frac1p}$.
For $c\neq 0$, consider instead the function $$g(x)=(px-c)\log(x)-\log(a)$$ $$g'(x)=-\frac{c}{x}+p \log (x)+p$$ $$g''(x)=\frac{c+p x}{x^2}$$ The first derivative cancels at $$x_*=\frac{c}{p\,W\left(\frac{e c}{p}\right)}$$ At this point $$g''(x_*)=\frac{p^2}{c} W\left(\frac{e c}{p}\right) \left(1+W\left(\frac{e c}{p}\right)\right)$$ Considering the value of $g(x_*)$ and the above would give you indications about the existence of the roots, their number and where they are located.