Craig Morgan and Ben Pringle have rejected new contracts at Rotherham and will leave the club on free transfers.
Saido Berahino can leave West Brom "if the deal is right", says manager Tony Pulis.
Clark's contract was terminated after an injury-hit second spell at Pittodrie while Magennis joined St Mirren on loan until the end of the season.
Logan, a 26-year-old right-back who is out of contract in the summer, has joined on a similar timescale.
The club confirmed that the signing of Logan, who could make his debut in Saturday's League Cup semi-final against St Johnstone, concluded their January transfer business.
Manager Derek McInnes told the club's official website: "He's got good experience and, while he's predominantly a right back, he can play at left back or in midfield as well.
"He also knows Nicky Weaver and Willo Flood from his Manchester City days and Niall McGinn from his time at Brentford."
Logan began his career at City, where he spent periods on loan at Grimsby, Scunthorpe, Stockport and Tranmere, and has made 85 appearances for Brentford since joining in 2011.
He said: "I first knew of Aberdeen's interest earlier in the week and I can't wait to get involved.
"There are so many big matches coming up and obviously I want to get playing as quickly as possible.
"The fans can expect a player who gives 110% in every game and I would describe myself as an attacking full-back."
McInnes gave Magennis hope of a return to Pittodrie after rejecting St Mirren's bid to sign the Northern Irish forward on a permanent basis.
McInnes said: "To be fair to Josh, he needs to playing regularly in order to have a chance of winning himself a new deal. We'll be monitoring his progress at St. Mirren and we certainly haven't closed the door on him."
But 33-year-old Clark has moved on after failing to force his way into the team since recovering from a knee injury.
Clark started his career with the Pittodrie outfit in 2000 before moving to Plymouth eight years later, returning to his home town club in 2011.
The Dons have now decided to release the player following 277 appearances and 13 goals for the club.
A statement read: "Aberdeen Football Club has terminated the contract of Chris Clark.
"A long-standing servant of the club, Chris has fully recovered from the injury he sustained in a pre-season game in Ireland but has been unable to force his way into first-team contention."