To monitor the exact time your buddy signs on and off while you are away, you need to enable the time stamp feature. First bring up your AIM Preferences window by clicking the preferences icon in the Buddy List toolbar

or by right clicking your system tray AIM icon.

Now select the "IM/Chat" category on the left hand side, and then check the box that says "Always view timestamp".

Return to your buddy list and double click the screen name of the buddy you would like to monitor.  This will bring up an empty Instant Message box, leave this box open while you are away and when you return, you will see the exact time your buddy signed on and off.