When you update Drupal core or any contributed module, it's a good idea to review the Recent log messages page (admin/reports/dblog) a few hours after the update. You want to make sure that there are no major PHP errors being generated after a deployment.
You can also use the log page to see who has logged in, modified content or created a comment. Often when I have to fix a bug on a production site, the Recent log messages page is the first place where I start my investigations.
But keeping track of the recently generated logs can be difficult, this is where you should look at using the Logging and alerts module to setup email notifications.