In preliminaries, Chapter 2, we define a Banach Algebra, Spectrum, Resolvent Set, Spectral Radius, Homomorphism, Antihormomophism, Nilpotent and Quasinilpotent. Propositions with proofs are used to explain the concept of Quasinilpotent Equivalence and the book culminates in showing when Quasinilpotent implies equality by proving the theorem: Suppose that a and b in a Banach Algebra A are quasinilpotent equivalent and suppose that there is an entire function f which has simple zeros. If f(a) = f(b) = 0, then a = b.