The advice to keep soap use to none or the absolute minimum, and to keep it far from any water source is good advice. As for a completely "odorless" bar soap so that bears will not be attracted to it, keep in mind that a bear's sense of smell is at least ten times better than a bloodhound.

All bar soap will have some kind of lipids (fat) as its main ingredient and all of them will have a smell that a bear could detect very easily, no matter what our noses can detect. Any bear that is willing to enter your campsite will take the time to investigate your hanging soap.

Whether a bear will be enticed enough to enter your campsite will depend entirely upon how the bear feels about humans. Some bears are human-habituated and do not associate us with danger, but with FREE EATS! Other bears think humans are to be avoided at all costs. Most of this will depend on the history between humans and bears in the location where you camp. The soap itself is unlikely to be a decisive factor.