Logarithm of an element near the identity
= Logarithm of an element near the identity
{title2=$\log(1-x)=-\sum_{n\ge1}x^n/n,\quad\|x\|<1$}
The displayed series converges absolutely in a <Banach algebra>. It is the <holomorphic functional calculus> value of the analytic branch $z\mapsto\log(1-z)$ on the unit disk. The <composition rule for holomorphic functional calculus> and $\exp(\log(1-z))=1-z$ give $\exp(\log(1-x))=1-x$. Thus every sufficiently small norm perturbation of the identity is an exponential.