Problem 2 (2020 AIME I)

There is a unique positive real number $x$ such that the three numbers $\log_8(2x)$, $\log_4(x)$, and $\log_2(x)$, in that order, form a geometric progression with positive common ratio. The number $x$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.