The Captain's Log voiceovers soon became simply a device for exposition, backstory, and summarizing the plot thus far. There were a few episodes where Kirk made a "log" entry while away from the ship without a tricorder.
I think I also recall a few log entries where Kirk states facts he couldn't possibly know, simply to inform the audience what's happening after a commercial.
That is true. In "The Enemy Within" I think it was called (where Kirk was split into good and evil parts by the transporter) Kirks logs refers to what has happened long before he or anyone else actually knows about the accident.
It is obvious that while some log entries were "real time observations" that others seemed to have been done much later and represented a "sumation" of the events that happened.
If you look at the Captain's Log entries as more of a personal diary rather than an official presentation of events then it makes more sense.
I believe he refers to unknown things in The Man Trap and Mudd's Women(?) as well. Regardless, that was a practice that was dropped after just a few episodes. If one really wants to justify you can just imagine that he was recording those logs after the fact.