As stated above, a good 140 gr bullet in the vitals will work just fine. And again, no match ammo or target bullets; you want reliable penetration and expansion.
You did not state the type of hunting you'll be doing.
Stand hunting over a bait, where shot distances are short, may not be the ideal cartridge choice, but will still work. Just beware the shot angle, and try to ensure the bullet is exiting low in the chest/shoulder on the off side to assist in a good blood trail.
On a level broadside or quartering away shot, aim for the bottom third of the chest to allow blood to drain and not collect inside the blood trail; again to promote a good blood trail.
Ideally, try to break the offside shoulder with your shot, as it will assist on knocking the bear down, and hopefully allowing for a quick follow up shot, if required.
Main thing here, is to reload quickly and be prepared for a quick follow up shot...do NOT sit there admiring your first shot, or you will miss an opportunity if needed!