This is the official manual for the latest Org-mode release.
The most basic logging is to keep track of when a certain TODO item was finished. This is achieved with1
(setq org-log-done 'time)
Then each time you turn an entry from a TODO (not-done) state into any of the
DONE states, a line ‘CLOSED: [timestamp]’ will be inserted just after
the headline. If you turn the entry back into a TODO item through further
state cycling, that line will be removed again. If you turn the entry back
to a non-TODO state (by pressing <C-c C-t SPC> for example), that line
will also be removed, unless you set
nil. If you want to record a note along with the timestamp,
(setq org-log-done 'note)
You will then be prompted for a note, and that note will be stored below the entry with a ‘Closing Note’ heading.