# Set var to directory which needs to be cleaned up
LOGDIR=/var/log/app/logs

cd $LOGDIR
find . -type f -ctime +28 -name 'applogfile.log*' -exec rm {} \;
find . -type f -ctime +28 -name 'app2logfile.log*' -exec rm {} \;

Note: this cleanup script removes files more than 4 weeks ago and have a name that starts with applogfile.log or app2logfile.log.

Other options are -mtime, for files that were modified more than 4 weeks ago.