To outlast something is to go on longer than it does — to still be there, still working, or still in use after the other thing has stopped. A stone house can outlast a wooden one. One battery can outlast another. A song can outlast the summer it was famous in. The idea is always a race against time, not a race of speed: you win by not finishing first, but by still being around. It is an ordinary, useful verb, fine in speech and in writing. Though actually, people sometimes stretch it a little: you can 'outlast a storm' meaning you stay until the storm is over, as if the storm were the thing you were competing with. That is close to the core sense, just less clearly 'longer than the other object'. Outlive is the sister word for living things: a person outlives another person. Outlast is the wider one — things, feelings, and fashions can outlast too. Last here is the verb 'to last' (to go on for a time), not last as in 'the last page'.