One way to write to the logs in Airflow is to return a string from a PythonOperator like on line 44 here.
Are there other ways that allow me to write to the airflow log files? I've found that print statements are not saved to the logs.
Answer
You can import the logging module into your code and write to logs that way
import logging
logging.info('Hello')
Here are some more options
import logging
logging.debug('This is a debug message')
logging.info('This is an info message')
logging.warning('This is a warning message')
logging.error('This is an error message')
logging.critical('This is a critical message')