By default 10.0 and higher delete all log files older than 15 days -- up until 10.2 M020, at which point log files are gzipped after 15 days and deleted only after 31 days.
These thresholds are configurable via the LogsDirViewer MBean in question in the server manager's JMX configuration. This cleanup can also be disabled entirely there.
One can configure a custom delegate that specifies what automated action should be done to old log files if you have something else in mind, e.g. moving the file to another folder. Unless you're doing something really fancy such a delegate should require very few lines of Java code. The out-of-the-box deletion and compression actions are nothing more than default delegates.