Yes, there are onboard computers on your car, but NO,none of them can log things like when the engine was last started or for how long. This apply to every vehicle having regular onboard computers.
Those computers are strictly designed to manage the various parts work and interactions. They have a good amount of ROM containing all programms and a small amount of RAM, just enough to work, a few flash error state registers, and NO long term storage at all, except, maybe, some prety small to remember seats position and some other user settings.
Kuälinar
February 2012