See:
https://en.wikipedia.org/wiki/Chip_log
https://en.wikipedia.org/wiki/Logbook_(nautical)
In the modern computing context, "logging in" does involve adding entries to a kind of (digital) record-keeping tool.
Did you log in (space: split verb)? What is your login (no space: noun)?
Do you have a backup (noun)? Did you back up (verb)?
Um, there is, actually.
On the etymological front, "log" was originally an actual wooden log, dragged behind a ship. It then became the book in which measurements of the log's speed were entered.
The UCLA Computer Club had a log -- a book with dated entries. Someone augmented that with a wooden log ... an intricate culture developed around this pairing. Traces of this are preserved: https://americanhistory.si.edu/collections/object/nmah_69051...
(That drawing is a pretty good representation of the actual log. And there's another memorialized pun: MUS stood for machine unit second, which was the computing center measurement unit -- the Computer Club had a quota for submitted jobs [card decks].)