2015 AIME II Problem 4

In an isosceles trapezoid, the parallel bases have lengths $\log3$ and $\log192$, and the altitude to these bases has length $\log16$. The perimeter of the trapezoid can be written in the form $\log2^p3^q$, where $p$ and $q$ are positive integers. Find $p+q$.

Leading zeroes must be inputted, so if your answer is 34, then input 034. Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.