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 want to record a note along with the timestamp, use2
(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.