We have an established logging system for our server-side services. Specifically, our Django project makes heavy use of the Python logging module, so call calls to logger.
logger.
Here is another question from here on SO about this very issue.
Some recommendations are: log4js, log4javascript, and Blackbird.
FWIW, log4js was the accepted answer there. I don't have experience with any of these platforms, so I can't really recommend one over the other.