Gaudin's agreement is worth $2.95 million plus possible incentives, and Logan signed for $590,000.
Gaudin, 26, came to the Yankees in a midseason trade with the Padres in 2009. Pitching both out of the rotation and in relief, Gaudin posted a 2-0 record and a 3.43 ERA as the Yankees won all six games that he started.
Owning a career record of 34-35 with a 4.50 ERA in 216 Major League appearances (75 starts) with the Rays, Blue Jays, Athletics, Cubs, Padres and Yankees, Gaudin should be presented with an opportunity to remain in New York's rotation.
He will enter Spring Training competing for the fifth starter's job along with contenders like Joba Chamberlain, Phil Hughes, Sergio Mitre and Alfredo Aceves, and could also be considered as a long reliever.
Logan, 25, came to the Yankees from the Braves as part of the Javier Vazquez deal in December.
A left-hander who owns 164 career Major League appearances with the White Sox and Braves, going 5-5 with a 5.78 ERA, Logan figures to be considered as the second southpaw in the Yankees bullpen behind Damaso Marte. Logan could assume the role last filled by Phil Coke, who was dealt to the Tigers in a December three-way trade that brought Curtis Granderson to New York.
The Yankees also recently agreed to terms with right-hander Sergio Mitre, avoiding arbitration. New York had one additional player eligible for salary arbitration this season, but instead chose to non-tender Chien-Ming Wang, cutting ties with the right-hander altogether.
Bryan Hoch is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.