How to find and read boot up log files?
Unfortunately for you, the Android log files use a "ring buffer" (think of it as an Ouroboros), so you might not be able to see the startup logs anymore. But here are some things you can do:


Check the "battery monitor" in Settings. Provided you didn't put your device to charge again since, it should show you how long it's been running without being attached to a charger
Using a terminal app or adb shell, run the command uptime – which tells you exactly how long the device has been running since it was last powered up (should also be listed in Settings › About › Status on your device)
different logs can be obtained via logcat, dmesg, and others (see our logging tag-wiki). On ICS (Android 4.0.x), you also can use Logcat &amp; Co..


Keep in mind that every action the device performs "scrolls on" the ring buffer, so older events get flushed out; installing an app will do so, but the other actions as well: screen tapped/swiped etc. are also events. I'd suggest starting with Settings › About › Status if you're after the time your device was robbed of its card.

As for the modified files: At least the Playstore app periodically checks things in background (not sure about the Youtube app here).