Never did find the answer to this, but I made my own alternative.
I created rc.local_jjp in /etc
#!/bin/sh -e
#
# rc.local_jjp
#
# This script is executed at the start of runlevel 0 and 6
# and at the end of runlevels 1,2 and 3
# Make sure that the script will "exit 0" on success or any other
# value on error.
#
LOG='/var/log/maint/mysyslog'
DATE="`date`"
case "$1" in
start)
start
;;
stop)
echo "System Shutdown at $DATE" >> $LOG
;;
restart)
echo "System Restart at $DATE" >> $LOG
;;
*)
echo "Usage: $0 {start|stop|restart}" >> $LOG
exit 1
esac
exit 0
Then, I symlinked it into the runlevel directories.
sudo ln -s /etc/rc.local_jjp /etc/rc0.d/K02rc.local_jjp
sudo ln -s /etc/rc.local_jjp /etc/rc1.d/S99rc.local_jjp
sudo ln -s /etc/rc.local_jjp /etc/rc2.d/S99rc.local_jjp
sudo ln -s /etc/rc.local_jjp /etc/rc3.d/S99rc.local_jjp
sudo ln -s /etc/rc.local_jjp /etc/rc6.d/K02rc.local_jjp
(I wasn't really clear on what runlevels 4 and 5 do, so I didn't add symlinks for them.)
Now, I have my own little syslog which shows startups and shutdowns.
I won't say much about Linux startup and shutdown behavior (mainly because I don't understand it that well), but the basic idea is that scripts in special directories get executed when the system changes runlevel such as when it changes from boot up to multiuser (runlevel 3).
Since my script needs the system to be running normally and because I don't want to chance interfering with something important, I named the symlinks so that it would get executed as one of the last things on startup and one of the first things on shutdown.