Logan joined the Scottish Premiership outfit on loan in January and went on to make 19 appearances, including their triumph in the League Cup final.
The 26-year-old – who came through the ranks at Manchester City – had been at Brentford since 2011, but will join Aberdeen in July on a two-year deal.
Aberdeen finished third last term and will enter the qualifying rounds for the UEFA Europa League, which helped convince Logan that the club was the right fit for him.
"The main reason I signed was because of how much I enjoyed it last season," Logan told RedTV .
"After two-and-a-half years at Brentford I was not really involved so it was great that I got an opportunity to play week in week out.
"European football was also a factor. When I was with Manchester City I was on the bench for a European game against Hamburg. That was a great experience so I am looking forward to hopefully playing this time."
Aberdeen manager Derek McInnes told the club's official website. "We're really pleased to get Shay.
"He will no longer be a loan player and will be an Aberdeen player and I think that's really important.
"I thought he showed enough in his loan spell with us last season in terms of his qualities and we're just looking for that consistency of performance going forward."
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