Logue was optioned to Triple-A Las Vegas on Sunday.
Logue started Saturday and picked up the win after giving up two earned runs over five innings, but he'll head back to the minors since the A's won't need a fifth starter until after the All-Star break. The left-hander has a 5.16 ERA, 1.45 WHIP and 23:11 K:BB across 29.2 big-league innings this season.