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log has 2 distinct synonyms spread across 2 of its 7 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
the exponent required to produce a given number
Synonyms, antonyms and the broader and narrower relations for log come from Princeton University’s WordNet. Full sources and licences.
a written record of messages sent or received
“they kept a log of all transmission by the radio station”
a written record of events on a voyage (of a ship or plane)
measuring instrument that consists of a float that trails from a ship by a knotted line in order to measure the ship's speed through the water