jQuery Client Side Logging

This plugin allows you to store front end log/info/error messages on the server (note: you will need to have something set up on your server to handle the actual logging).

jQuery Error Logger

A simple jQuery logging plugin to log exceptions and send them to your API instantly or periodically. If you opt for the periodic option it will store the errors in localstorage and then post them to your server based on the interval you choose.

JLoggins

jQuery plugin for console logging