Rather than reboot the whole server, I wanted to restart the service that writes to auth.log. I found a reference to rsyslog, but I was a couple of versions of Ubuntu later than the 10.04 mentioned in that conversation. (Ubuntu 12.04 LTS) so I ended up getting "rsyslog: unrecognized service" when I tried that.
I ran "service --status-all" to get a list of services on the machine and "sysklogd" looked promising so I tried restarting it.
user@server:~# service sysklogd restart * Restarting system log daemon... [ OK ]
Then I exited SSH and logged back in, to find that my auth.log file was getting written to again.