Without being able to see that section of your auth.pl file it's hard to tell where a mistake may be.
be sure at the end of the following line, you have \n";
print LOG "$uid $action at $time on $date from $host\n";
If that doesn't work perhaps you can post a .txt copy of your auth.pl file and provide the url to where it can be viewed.
Unoffical DBMan FAQ
http://creativecomputingweb.com/dbman/index.shtml/
be sure at the end of the following line, you have \n";
print LOG "$uid $action at $time on $date from $host\n";
If that doesn't work perhaps you can post a .txt copy of your auth.pl file and provide the url to where it can be viewed.
Unoffical DBMan FAQ
http://creativecomputingweb.com/dbman/index.shtml/