Coach Sonny Dykes announced the news on Saturday before the Golden Bears (1-3, 0-1 Pac-12) hosted Washington State.
Adcock injured his knee in practice on Wednesday. He started the first four games at center and is expected to be replaced by Mark Brazinski.
Logan started two of the first four games, recording 11 tackles and intercepting a pass. He has dealt with a variety of injuries in his career leading to his decision. Logan is on track to graduate in December.
Cal kicked defensive end Chris McCain off the team earlier this week for conduct detrimental to the team.