How to log error to file, and not fail on exception

You could use the logging module:

import logging
logging.basicConfig(filename="/tmp/myapp.log", level=logging.DEBUG, 
                    format="%(asctime)s %(levelname)s %(name)s %(message)s")
logger=logging.getLogger(__name__)

try:
    1/0
except ZeroDivisionError as err:
    logger.error(err)

Running the script writes in /tmp/myapp.log:

% cat /tmp/myapp.log 
2010-08-01 17:50:45,960 ERROR __main__ integer division or modulo by zero