A bear is one animal that can be dropped easily when shot from behind, just aim at the root of the tail and your bullet will break the spine. When the animal is quartering the technique that I've discovered working quite well is to aim at the space between the front legs, come up 1/3 of the way up the body and fire. This hold prevent the bullet from hitting to far back or too far forward. If you are shooting from above - such as in a tree stand or from a rock out crop, aim behind the shoulder rather than on top of the shoulder as the heavy bones might cause a standard bullet to fail, a premium should hold together. From the excellent pic supplied by Goose25, we see that there is little for a bullet to hit in the top half of the chest, unless you luck out and hit the spine. Head on, if the bear's head is low aim forward of the shoulder and shoot him through the top of the neck, if the bullet misses the spine it will continue into the chest. If the bear's head is up, center the chest.
IMHO, it to be a mistake to consider the black bear not dangerous. If you are offered a shot that will drop the bear rather than letting him run into the thick stuff, take it. If you need to follow up be alert! When you approach your "dead " bear, don't make any assumptions, if your not sure, shoot him again. No, never mind that - just shoot him again anyway.