1960 AMC 12 Problem 24

Convert the logarithmic equation to $(2x)^x=216$

Use $216=6^3$ to recognize a real solution

The equation is equivalent to $$(2x)^x=216.$$ Since $216=6^3,$ $x=3$ satisfies the equation. To see it is the only admissible real solution, note that the base requires $x>0$ and $2x\ne1;$ on $0\lt x\lt\frac{1}{2},$ $x\ln(2x)\lt0,$ while for $x>\frac{1}{2}$ the function $x\ln(2x)$ is strictly increasing. Thus $x=3,$ a non-square, non-cube integer.

Therefore, the correct answer is A.