That said turning on logging is easy enough when you find the option: fire up the Task Scheduler, right-click on the Task Scheduler Library, and pick Enable All Tasks History.
Anyway, the culprit is the mcupdate_scheduled task, buried away in the Task Scheduler Library under Microsoft → Windows → Media Center. It seems to automatically reschedule itself every so often, and has the "Wake the computer to run this task" option ticked. So if it decides to next run at, say, 3:59am on Wednesdays it'll happily fire the computer up in the wee hours of the morning. Fortunately there's a nice easy fix: open the Properties, and on the Conditions tab untick the "Wake the computer to run this task" option.