You can use any log you want, but the heavier the wood, the more "meat" there is for the fungus to consume and the longer growing lifespan you'll get out of it.
An oak log should grow in production volume (harvest every 5 weeks) for 4-5 years after first fruit (after the incubation period). Aspen might rot out in the first couple years because the bark isn't thick enough to protect it from other predation (micro organisms that want to eat it too), and it's much less dense and thus less food for the shrooms to work with.
You want a reasonably fresh log because of the moisture content still retained in the live wood, and because it hasn't already been colonized by other mushrooms. Do you want a fresh hamburger right off the stove, or the one that's been sitting on the counter at the check out at a truck stop for the past 7 months? :)