(covers information from several alternate timelines)
The captain's log was a form of log entry record-keeping that was used since the first captains sailed Earth's seas in ancient history. The log was used to inform the captain's superiors of what was happening on a mission and to record historical facts for future generations.
In the 2150s, the United Earth Starfleet simply used the day, month, and year of the Earth Gregorian Calendar to record dates in starlogs. The captain's log was known in this period as the captain's starlog, while an acting captain would keep an acting captain's starlog. (ENT: "Broken Bow", "Hatchery")
When breaking and entering a closed curio shop on the Akaali homeworld as part of a covert mission in 2151, Trip Tucker suggested that Captain Jonathan Archer shouldn't mention that in his log. (ENT: "Civilization")
- Military log
- Personal log
- Captain's Log: William Shatner's Personal Account of the Making of Star Trek V: The Final Frontier
- Star Trek: Fan Collective - Captain's Log
Obviously, the "real" purpose of the log is to inform the audience about key plot points; it should be noted that in some of the first episodes of the original series, Kirk's log sometimes broke the fourth wall and described information that neither he nor the crew could be aware of. For example, in "The Naked Time", Kirk describes "… but unknown to us…". There clearly wasn't time to make some other log entries; for instance, in "By Any Other Name", Kirk appears to have made a log entry about his crew's skulduggery against the episode's villain while he, Kirk, is sitting next to that same villain.
Presumably, these log entries were created after the events had transpired, when the captain had time to update his log, and it was either Starfleet procedure or just Kirk's personal habit to record such logs in the present tense.