Utah Code § 9-7-503

(1) Each director of a library board shall be appointed for a four-year term, or until the director's successor is appointed. Initially, appointments shall be made for one-, two-, three-, and four-year terms, and one member of the county legislative body for the term of his elected office. Annually thereafter, the county executive body shall, before the first day of July of each year, appoint, with the advice and consent of the county legislative body, for a four-year term, one director to take the place of the retiring director. (2) Directors shall serve not more than two consecutive full terms. (3) The directors shall annually select a chairman and other officers. (4) The county executive body may remove any director for misconduct or neglect of duty. (5) Vacancies in a library board of directors shall be filled for the unexpired terms in the same manner as original appointments.