Is there a log file I can check to tell me when the computer has been rebooted?
Use the event viewer to examine the event log.
Specifically you can filter the System event log for source Eventlog.
Event 6009 is logged at startup. It will look similar to the following:
“Microsoft ® Windows NT 4.0 1381 Service Pack 6 Multiprocessor free” It should be followed by Event 6005.
Previous to 6009, you should have one of these: Event 6006 is a clean shutdown of the system, 6008 is dirty.
ETA: I should have added that these events are only logged in SP4 and later, IIRC.