The 33-year-old has signed a three-year-deal with the ambitious League One side, managed by Paul Hartley
He becomes the third signing from the Premiership following the arrivals of Ross Draper and Iain Vigurs from Ross County.
Logan spent part of last season on loan at Hearts, adding experience to the defence as the club won the Championship. He kept a clean sheet in all five games he played.
The right-back joined Aberdeen in 2014 and would make 276 appearances across seven-and-a-half seasons, winning the League Cup and helping the Dons into Europe on multiple occasions.
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He could make his debut for Cove on Saturday when they face Falkirk with the teams expected to be rivals it out at the top of League One this season.
“We’ve been working on getting Shay over the last few weeks because it was clear he would be a huge asset to us,” Hartley told the club's website.
“It’s pleasing a player who was such a mainstay at Aberdeen for a number of years sees his future with us.
“Shay is the third player to join us this summer who was a regular at Premiership level last season, joining Iain Vigurs and Ross Draper.
“Their experience will be of massive benefit to the younger players at the club in what we know is going to be another very challenging League One campaign.”
Cove have also added Jevan Anderson, the son of former Aberdeen captain Russell, following his release from Burton Albion.
He has signed a one-year-deal with an option for another.