Linux: Send email every time there is a failed login

Check out the login failure daemon which is a part of
http://configserver.com/cp/csf.html

Which syslog us in use on your system?  Changing to rsyslog would enable you to configure it when an event such as a failed login is received, it will send out an email.  

I would advise against using an email client such as mail, etc.  a simple shell script that directly injects the message into the nail server queue.

How can I find out what syslog is in use?

Which Linux is running?
ps -ef | grep syslog

ls -l /usr/sbin/syslogd

I use CentOS.

[root@server]ps -ef | grep syslog
root      1050 15809  0 16:56 pts/4    00:00:00 grep syslog
root      2303     1  0 Nov15 ?        00:00:19 syslogd -m 0

[root@server]ls -l /usr/sbin/syslogd
ls: /usr/sbin/syslogd: No such file or directory

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I understand your point about mail being a bad idea.  Is there a way to configure syslog to just write it to a log file?  "lastb" is unreliable.  It shows no new entries since Monday.