As far as client side logging is concerned there is no mechanism as per my knowledge.
Because the developer never maintain any database of stuff on client side. Even if you save exception in client side storage that wont be a centralized approach and how will you be getting back again?
What you can do
By client side you can log javascript's specific scenarios/Exceptions via Ajax Calls to your server in the same error logging module that you are using for server error logs and saving those exception you want to log in your database to have their centralized record.