Mich. Comp. Laws § 397.203

Sec. 3. The offices of boards of directors appointed before July 1, 1932 under this act, consisting of 9 members, are vacant effective July 1, 1932, and a board of 5 members to succeed them or a board of 5 members for a library newly established under this act shall be first appointed as follows: 1 member shall be appointed for a term of 5 years, 1 member shall be appointed for a term of 4 years, 1 member shall be appointed for a term of 3 years, 1 member shall be appointed for a term of 2 years, 1 member shall be appointed for a term of 1 year, and annually thereafter the mayor shall appoint 1 member for a term of 5 years. The mayor may, by and with the consent of the city council, remove any member for misconduct or neglect of duty.