I am looking trying to find a mail server to install on my servers, the only thing I want it to do is send the daily logs and any outgoing mail that php scripts would make (contact forms, notifications, etc) I use qmail for my main mail servers, I don't like the sendmail mess. I don't need qmail the server will only send about 5 messages a day, I also would like it to only listen on localhost, there is no incoming mail to it and the only address allowed to send through it is 127.0.0.1

Is there a very small mail server that could be recommended?

you can try using postfix.