You could just .split it:

>>> Path('logs/date.log.txt').stem.split('.')[0]
'date'

os.path works just as well:

>>> os.path.basename('logs/date.log.txt').split('.')[0]
'date'

It passes all of the tests:

In [11]: all(Path(k).stem.split('.')[0] == v for k, v in {
   ....:     'a': 'a',
   ....:     'a.txt': 'a',
   ....:     'archive.tar.gz': 'archive',
   ....:     'directory/file': 'file',
   ....:     'd.x.y.z/f.a.b.c': 'f',
   ....:     'logs/date.log.txt': 'date'
   ....: }.items())
Out[11]: True