Part of the problem will lie with the rope you use. A laid rope (spiral twists) will tend to unwind under load. Use a kernmantle (sheathed) rope to minimize that issue.

I would try a bowline on a bight -one loop expanded to envelop the log and brought back to the second loop (you can do as many loops as you desire, depending on the length of the rope you have and the size of the log), attaching to the smaller loop with a carabiner(s).

This works for large mammoth skulls - ought to work for logs.

Interesting problem! What other solutions are there?? - I'll bet Tony E has a dandy....