Isolate x in this equation

Question

I'd greatly appreciate it if someone could please isolate "x" by manipulating the following equation: $$(2^xR)+x=(x-1)p$$

Answer 1

This is a transcendental equation, so it can't be solved by pure algebra techniques; there's no closed form solution.

Answer 2

There is an explicit solution for $x$ using Lambert $W$ function. The solution is given by $$x=\frac{p}{p-1}-\frac{W\left(-\frac{2^{\frac{p}{p-1}} R \log (2)}{p-1}\right)}{\log (2)}$$ In the case where the argument of the Lambert $W$ function is small or large, there are very nice approximations which at least would give you a reasonable estimate of the solution for starting a Newton procedure.

In fact, any equation which can be rewritten as $A+Bx+C\log(D+Ex)=0$ has an explicit solution which involves Lambert $W$ function.