I'm not sure about your method, honestly I didn't really review it (once you started taking the logs I zoned out).
But here's what I would do. Take the derivative of your equation with respect to time = t;
Using product rule gets you:
Solve for V' to get the rate the volume is increasing.
The problem gives you P, P' and V, so you are all set to substitute in the values and get the answer. I got .