I have a 2009 Mac Mini running OS X 10.11.6 set up as home theater. Its fan has been quite noisy for a while, but it has still worked fine... until a couple of days ago, when I found out that it had shut down on its own. I turned the Mac again, the fan started making noise... and a few hours later it also shut down on its own (I don't know what message or pop-up it showed, because I have it plugged into my TV, and I didn't have it turned on).
My theory is that the fan couldn't cool the system well enough, so the CPU temp rose so much that OS X shut the computer down on its own, but when I turned it on today and looked on
/var/log/system.log, I couldn't find any message related to that, nor in any of the other log files available at the Console app.
Where would OS X store logs of such a thing happening, if any?