I would recommend giving Trackjs a try.
It's error logging as a service.
It's amazingly simple to set up. Just add one <script> line to each page and that's it. This also means it will be amazingly simple to remove if you decide you don't like it.
There are other services like Sentry (which is open-source if you can host your own server), but it doesn't do what Trackjs does. Trackjs records the user's interaction between their browser and your webserver so that you can actually trace the user steps that led to the error, as opposed to just a file and line number reference (and maybe stack trace).
