Nothing of great value, but the easiest way to log both to a file and to the console seems to be something like this:

import logging
#more verbose formatting for the logfile
logging.basicConfig(filename="blender-bundler.log", format="%(asctime)s %(message)s", level=logging.DEBUG)
logging.getLogger().addHandler(logging.StreamHandler())

#example:
logging.info("Some info here")