Is there a way to see how long is an extension alive for?
You could grep the “full” logs for when an extension was last “Reachable”
EXT=1010;cat /var/log/asterisk/full*|grep -i reachable|grep $EXT |sort -n
From my experience dealing with customers, you probably want a lot more than just “when did the extension last log in.”
If that’s the case, then “No” is the short answer. Could you find that information out? Yes, Is there a way to do it right now out-of-the-box? Nope.
If all you really want is a system that tracks when phones are and are not reachable, something like Nagios could be made to work. You can, as @dicko pointed out, write a series of grep/php scripts that drag through the logs to find whatever information you are looking for.