The 26-year-old full-back was given the punishment by referee Alan Muir for using 'offensive and insulting language' following the Dons' 2-1 loss.
Logan will serve a one-game ban meaning he'll miss the trip to Partick Thistle on November 23rd.
McInnes said: "Shay should have kept his cool. It was something he said in frustration to the linesman in the tunnel. They took offence to it and they have red-carded him.
"He, like the rest of my players, is a bit aggrieved at a couple of decisions in the build-up to the goals."