Exercise 2, § 4

Let $A$ be a unital Banach algebra and $G$ the group of invertible elements of $A$.

a) For an element $x\in A$ to be of the form exp($y$), where $y\in A$, it is necessary and sufficient that it be contained in a connected commutative subgroup of $G$.

b) For an element $x\in A$ to be of the form exp($y$), where $y\in A$, it is sufficient that 0 belong to the unbounded connected component of $\mathbf{C}-$ Sp$_A(x)$.