AMC 12 2023a Problem 6

Points $A$ and $B$ lie on the graph of $y=\log_{2}x$. The midpoint of $\overline{AB}$ is $(6, 2)$. What is the positive difference between the $x$-coordinates of $A$ and $B$?

\textbf{(A)}~2\sqrt{11} \textbf{(B)}~4\sqrt{3} \textbf{(C)}~8 \textbf{(D)}~4\sqrt{5} \textbf{(E)}~9