  Let $A$ be a complex Banach algebra. If the spectrum of an invertible element
$a\in A$ does not separate the plane, then $a$ admits a logarithm. We present
two elementary proofs of this classical result which are independent of the
holomorphic functional calculus. We also discuss the case of real Banach
algebras. As applications, we obtain simple proofs that every invertible matrix
over $\mathbb C$ has a logarithm and that every real matrix $M$ in $M_n(\mathbb
R)$ with $\det M>0$ is a product of two real exponential matrices.
