There are many Methods how to handle logs in Docker Instances:
Some of em:
BAD: Running a Log Collector like fluentd inside the container
BAD: Processing Logs Only in the Docker Container it self
GOOD: Processing logs on the Host System where Docker Deamon Runs on!
How to do it?:
Easy going /var/lib/docker/containers/hash/id/ <- there are the logs stored you can rotate and parse em directly from there and you can Adjust the Performance of the Docker Containers for Log Processing so that all runs Smoth Soon a Complet Tutorial this is only to get you started and how to do logging.