About the second equation, it seems a solution exists for a unique positive $x$.
Solutions cannot be negative, since the left hand side would be bigger than $2^n$.
On $\mathbb{R}_+$, whe have:
- The left hand side is a sum of increasing functions, so it is itself increasing.
- Letting $x$ go to $0$, the left hand side goes to $0$.
- Letting $x$ go to infinity, the left hand side goes to $2^n > 1$.
So there is a unique value in between that solves the equation, but I don't see how to solve it analytically.
Given its monotonic nature, it should be very easy to solve numerically, with a Newton-Raphson method, for example.