If a Moodle instance had a large number of users with a lots of activity, would this mean that the site logs would become very large? And could this affect performance?
If so, is there a way to limit the amount of information recorded?
Well, after a while this will hit performance for sure, as logs are stored in database. To limit log lifetime you can adjust settings in Administration -> Server -> Cleanup.
If you need to store those log for some reason, from performance point of view it would be better to back them up and clean the table.
If you need to store those log for some reason, from performance point of view it would be better to back them up and clean the table.