2007 AMC 12A Problem 23

Let $A=(p,\log_a p)$ and $B=(q,2\log_a q).$ Since $AB$ is horizontal, $\log_a p=2\log_a q=\log_a q^2,$ so $p=q^2.$ The side length is $6=|p-q|=|q^2-q|,$ whose only positive solution is $q=3.$ Since $C=(q,3\log_a q),$ the vertical side gives $BC=6=\log_a q=\log_a 3.$ Thus $a^6=3,$ so $a=\sqrt[6]{3}.$ Thus, the correct answer is A.