Let A and B be unital commutative Banach algebras. Suppose that A is semi-simple. Let ρ : A → A and τ : B → B be bijections. If T : A → B is a surjection with, for some α ∈ C \\ {0}, r ( T (f) r ( T ( s ) ) - α) = r ( f ρ( g ) - α) for all f,g ∈ A, then B is semi-simple and r ( T ( f ) T ( g )-1 - 1) = r(fg-1 - 1) for every f ∈ A and G ∈ A-1. As a consequence, T ( 1 ) is invertible and T ( 1 )-1 T is a real-algebra isomorphism. If, in addition, T(1)-1T(i) = i, then T ( 1 )T is a complexalgebra isomorphism. This result unifies and generalizes [3, Theorem 7.4] and [4, Theorem 3.2 and 6.2]. Copyright © 2011 Rocky Mountain Mathematics Consortium.