The right way to achieve what you ask is to use Logger object. It gives you much more flexability. This object can be bound to multiple handlers; You need a streamhandler to log message to sys.stdout and a file handler to log it to a file. You then both print to the screen and log to a file in a single command.
import logging
# create logger 
logger = logging.getLogger('example')
logger.setLevel(logging.INFO)
# create file handler which logs messages
fh = logging.FileHandler('fh.log')
fh.setLevel(logging.DEBUG)
# create console handler to print to screen
ch = logging.StreamHandler()
ch.setLevel(logging.INFO)
# create formatter and add it to the handlers
formatter = logging.Formatter('%(asctime)s - %(name)s - %(levelname)s - %(message)s')
fh.setFormatter(formatter)
ch.setFormatter(formatter)
# add the handlers to the logger
logger.addHandler(fh)
logger.addHandler(ch)
Now every call to logger.info(msg) will be printed both to the screen, and written to fh.log file.
There is another way, where you can replace sys.stdout and sys.stderr stream objects. Create a class and customize it (original answer here):
import sys
class writer(object):
    _fh = None
    _orig_stdout = sys.stdout
   def __init__(self):
       _fh = open('logging.log', 'w')
    def write(self, data):
        fp.write(data)
        _orig_stdout.write(data)
    def flush():
        _orig_stdout.flush()
logger = writer()
sys.stdout = logger
sys.stderr = logger