Every OS is different, up to and including different flavors of and installations of the same flavor of Linux.

But if you are using a Unix (including Linux, *BSD, etc), somewhere under /var/log, you might find a file that the activity of various daemons. On my box, I have /var/log/daemon.log, which gives the time, process id and username for every ssh connection opened and closed, as well as ftp, etc.

If I were to write such a program, all I would have to do is parse /var/log/daemon.log line by line matching usernames and process ids for get the starting and opening times.

However, keep in mind that lots of log analysis programs have been written. You'd probably be better of finding one by googling on "linux log analysis programs written in perl".