Proposition 7.3.15.label Let $x$ be an element of a unital Banach algebra $A$. If $\spec{A}{x}\suf\curl{\lam\in \com:-\pi<\Im\lam<\pi}$ then
\begin{align*}\log\exp x=x\end{align*}
Proof. By Spectral Mapping 7.3.12
\begin{align*}\spec{A}{\exp x}=\exp\parens{\spec{A}{x}}\end{align*}
Furthermore $\spec{A}{x}\suf\curl{\lam\in \com:-\pi<\Im\lam<\pi}$ hence
\begin{align*}\spec{A}{\exp x}=\exp\parens{\spec{A}{x}}\suf\com\del(-\infty,0]\end{align*}
This shows that $\spec{A}{\exp x}\suf\text{dom(Log)}$ so $\log\in \holon{\spec{A}{\exp x}}{\com}$ and by Spectral Mapping 7.3.12 with $f=\exp, g=\log$ we have
\begin{align*}\log\exp x=x\end{align*}
$\square$
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