An art logo design is usually created for an art gallery brand identity, artist, sculptor and painter, etc. Understanding who the target audience is, and who the art logo design is for, is important for designing it.
An art gallery logo design for example needs to make an impression on art connoisseurs, sponsors and artists alike. It should give that elitist brand image in order to attract visitors to the gallery.
An art studio logo design on the other hand represents the art style of the individual artist(s). The logo image should reflect this in order to attract buyers of this type.