Logic is not an authority, but neither is it a servant. Logic can't relate to you like a person, because it doesn't have ends, loyalties or passions. Logic cannot dictate what you should believe nor enforce any decrees; neither will it strive to please you or follow commands. Logic is an authority or servant only in a metaphorical sense; it is "ultimate" only with respect to the purely formal properties of truth and inference.
We can be wrong about logic, and we can err in our reasoning. Some of us just do not care about truth or reason. To assert the 'ultimate authority' of logic is rather pointless, since it does nothing to settle disagreements of fact or ends. If we accept an argument as logical, and we are interested in truth and reason, then we, not logic, are the authority who makes that decision, and we, not logic, are responsible for any errors that follow.
Logic cannot compel; it is not an actor. Our passions may be compelling, in some sense, and we may have a passion for truth and reason. A logical argument may seem "compelling", but such compulsion does not come from logic itself, but rather from our passion for truth and the logic of argument.
Logic has no authority to compel unless we surrender ourselves to it, but then we are merely giving unchecked power to our fallible attempts to reason. It's best, I think, to eschew the idea of an authority in such matters. It is the parochial delusion of a species preoccupied with social strata: a crude attempt to shoehorn an abstraction into the place of a hero or leader. I love logic, but not this "authority" that people try to make of it.