How can I support a spinning log for a playground set?
You're going to need bearings. Even if you drill precisely and grease the pins, there will be substantial drag over time. The weight of small children won't be enough to rotate the logs. 

I'd look into something like these tray bearings. Fit them between the end of the logs and your posts. It'll take some strategery to get it all together with limited space, but it can be done. 

Obviously you want bearings substantial enough for this duty. Household "lazy Susan" hardware won't cut it. Be aware that load ratings are for horizontal application and don't apply when installed vertically, where shear forces are much greater. 



If those don't work for some reason, roller bearings are readily available and inexpensive. They'd require you to accurately embed them in the logs and use a stud that fits well and is rigid enough to remain horizontal.