One more method is to wrap the logger in an object that translates calls to write to the logger's log method.
Ferry Boender does just this, provided under the GPL license in a post on his website. The code below is based on this but solves two issues with the original:
- The class doesn't implement the flush method which is called when the program exits.
- The class doesn't buffer the writes on newline as io.TextIOWrapper objects are supposed to which results in newlines at odd points.
import logging
import sys
class StreamToLogger(object):
    """
    Fake file-like stream object that redirects writes to a logger instance.
    """
    def __init__(self, logger, log_level=logging.INFO):
        self.logger = logger
        self.log_level = log_level
        self.linebuf = ''
    def write(self, buf):
        temp_linebuf = self.linebuf + buf
        self.linebuf = ''
        for line in temp_linebuf.splitlines(True):
            # From the io.TextIOWrapper docs:
            #   On output, if newline is None, any '\n' characters written
            #   are translated to the system default line separator.
            # By default sys.stdout.write() expects '\n' newlines and then
            # translates them so this is still cross platform.
            if line[-1] == '\n':
                self.logger.log(self.log_level, line.rstrip())
            else:
                self.linebuf += line
    def flush(self):
        if self.linebuf != '':
            self.logger.log(self.log_level, self.linebuf.rstrip())
        self.linebuf = ''
logging.basicConfig(
    level=logging.DEBUG,
    format='%(asctime)s:%(levelname)s:%(name)s:%(message)s',
    filename="out.log",
    filemode='a'
)
stdout_logger = logging.getLogger('STDOUT')
sl = StreamToLogger(stdout_logger, logging.INFO)
sys.stdout = sl
stderr_logger = logging.getLogger('STDERR')
sl = StreamToLogger(stderr_logger, logging.ERROR)
sys.stderr = sl
This allows you to easily route all output to a logger of your choice. If needed, you can save sys.stdout and/or sys.stderr as mentioned by others in this thread before replacing it if you need to restore it later.