Answer: 18

Small Hint:

Each leg is the hypotenuse of a right triangle with legs $\log 16$ and half of $\log 192 - \log 3$

Big Hint:

Those legs are $4\log 2$ and $3\log 2,$ so each slanted side is $5\log 2$ by a $3$-$4$-$5$ triangle

Solution:

Dropping altitudes from the ends of the short base, each leg is the hypotenuse of a right triangle whose legs are the altitude $\log 16 = 4\log 2$ and half the difference of the bases, $\frac{1}{2}(\log 192 - \log 3)$ $= \frac{1}{2}\log 64$ $= 3 \log 2.$ By the $3$-$4$-$5$ ratio, each leg has length $5 \log 2.$

The perimeter is $$\begin{aligned} &\log 3 + \log 192 + 2 \cdot 5\log 2 \\ &= \log(3 \cdot 192) + \log 2^{10} \\ &= \log(2^6 3^2) + \log 2^{10} \\ &= \log 2^{16} 3^2, \end{aligned}$$ so $p + q = 16 + 2 = 18.$