Who owns the rights to the artwork?
The law says the artist. Does this mean you cannot trademark or register your logo? Of course not, because that useage is what you specifically pay for. The art was created for you, and you paid for its use for that purpose. It is entirely yours to use. Legally, there is no difference between artists, whether graphic artists, musicians, photographers, or actors. If a rock band writes a commercial jingle for a product, they give the rights to use it for that purpose, not the ownership of the music for all future uses. It is no different for artists of any kind. You own the rights you paid for.