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1. The act of writing information to a log (see definition 2) or to a log file. The term originates from the practise of a maritime officer maintaining a record of events in a ship's log.
2. A record of what has happened, i.e. a record of events. Historically a written diary, such as a captains log recording events on a voyage. In this context it can also refer to a log file.
3. Abbreviation for logarithm, a mathematical function.