2021 Fall AMC 12A Problem 9

Let $a = \log_2 x,$ $b = \log_3 x,$ $c = \log_4 x.$ Surface area equals volume gives $2(ab + bc + ca) = abc.$ Dividing by $abc,$ $$1 = 2\left(\frac{1}{c} + \frac{1}{a} + \frac{1}{b}\right).$$ Since $\dfrac{1}{a} = \log_x 2,$ etc., the sum is $\log_x 2 + \log_x 3 + \log_x 4 = \log_x 24.$ Thus $1 = 2\log_x 24,$ so $\log_x 24 = \tfrac{1}{2},$ meaning $x^{\frac{1}{2}} = 24$ and $x = 576.$ Thus, the correct answer is E.