Your pretty much correct in that the cabin pressure is around 8000ft. However that is not until the aircraft is cruising at around FL390. At Fl 350 is it around 6000 ft. The pressure is reduced gradually from take off elevation to cruising altitude, it doesnt just maintain ambient external pressure to 8000 ft and then hold that. If that was the case your ears would really "pop" especially if the reciprocal were true on the descent.
Modern jets operate on a pressurisation schedule which carefully regulates the pressure during all phases of the climb and descent to provide smooth pressure change gradients. The whole subject does get fairly technical and involves the time an aircraft maintains a given altitude, the landing field elevation, the final cruise altitude, descent and climb modes etc. However this is a fairly simple explanation which should answer your question.