2021 Fall AMC 12A Problem 9

A right rectangular prism whose surface area and volume are numerically equal has edge lengths \log_{2}x, \log_{3}x, and \log_{4}x. What is x?

\textbf{(A)}\ 2\sqrt{6} \qquad\textbf{(B)}\ 6\sqrt{6} \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 48 \qquad\textbf{(E)}\ 576

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.