Yes, this is possible, but most likely not in the way you'd like to use it.
.tar.gz is actually two things in one: gz or gzip is being used for compression, but this tool can only compress single files, so if you want to zip multiple files to a compressed archive, you would need to join these files first. This is what tar does, it takes multiple files and joins them to an archive.
If you have a single long logfile, just gziping it would be easier. For this, Python has the gzip module, you can write directly into the compressed file:
import gzip
with gzip.open('logfile.gz', 'a') as log:
    # Needs to be a bytestring in Python 3
    log.write(b"I'm a log message.\n")
If you definitely need to write into a tar-archive, you can use Python's tarfile module. However, this module does not support appending to a file (mode 'a'), therefore a tarfile might not be the best solution for logging.