A log or log entry is a recording which is entered into a starships computer record for the purposes of posterity. Officers aboard United Earth and Federation starships are required to record official logs, which is denoted by their rank and position.
Starfleet logs typically noted the stardate at the beginning of the log for the purposes of ordering log recordings. Klingon Commander's log entries used a numerical designation instead of dating. For instance commander Kagh of the IKS Vortha's final log entry was number 5373. (TOS comic: "Against Their Nature")
In 2370, retired Starfleet Captain Montgomery Scott rendezvoused with the USS Enterprise-D in order to browse the logs of the original USS Enterprise crew, looking for Spock's time travel equations. (Star Trek novel: Engines of Destiny)