@pseudoramble My old gym had a couple logs, 8" and 12".
They were fun to play with, but didn't seem to have much use beyond log clean/press. If you're doing strongman that's super useful, but as I don't, they were just a novelty.
If I may, I'd recommend looking at an axle/square bar or a camber swiss bar. The latter was super helpful for shoulder rehab and I'd argue is the reason my bench has blown up.
Whatever you decide, I'm excited to hear about it!