Is there a closed form for the following sum, where $x>0$? $$\sum_{n=1}^{\infty} \frac{\ln(x+2^n)}{2^n}$$
The sum pops up when considering the expected utility of a game where one receives $\$2^n$ if the first head occurs on the $n^{th}$ flip of a fair coin, where the utility function is $U(w)=\ln (w)$. I am not interested in convergence proofs, only closed form expressions.
