$\newcommand{\+}{^{\dagger}}
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\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\downarrow}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,{\rm e}^{#1}\,}
\newcommand{\fermi}{\,{\rm f}}
\newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}
\newcommand{\half}{{1 \over 2}}
\newcommand{\ic}{{\rm i}}
\newcommand{\iff}{\Longleftrightarrow}
\newcommand{\imp}{\Longrightarrow}
\newcommand{\isdiv}{\,\left.\right\vert\,}
\newcommand{\ket}[1]{\left\vert #1\right\rangle}
\newcommand{\ol}[1]{\overline{#1}}
\newcommand{\pars}[1]{\left(\, #1 \,\right)}
\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
\newcommand{\pp}{{\cal P}}
\newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,}
\newcommand{\sech}{\,{\rm sech}}
\newcommand{\sgn}{\,{\rm sgn}}
\newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}}
\newcommand{\ul}[1]{\underline{#1}}
\newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}
\newcommand{\wt}[1]{\widetilde{#1}}$
$\ds{a \in {\mathbb C}}$.
Lets $\ds{z \equiv r\expo{\ic t}\ \imp\ \dd z = r\expo{\ic t}\ic\,\dd t\ \imp\dd t = {\dd z \over \ic z}}$:
\begin{align}
I&=\left.\int_{0}^{2\pi}\ln\pars{\verts{re^{\ic t} - a}}\,\dd t
\,\right\vert_{\,0\ <\ r\ <\ \verts{a}}\ =\
\Re\int_{0}^{2\pi}\ln\pars{re^{\ic t} - a}\,\dd t
\\[5mm] & =
\Re\oint_{0\ <\ \verts{z}\ =\ r\ <\ \verts{a}}
\ln\pars{z - a}\,{\dd z \over \ic z} =
\Re\pars{2\pi\ic\lim_{z\ \to\ 0}\bracks{z\,{\ln\pars{z - a} \over \ic z}}} =
2\pi\,\Re\pars{\ln\pars{-a}}
\\[5mm] &=
\bbox[10px,border:1px groove navy]{2\pi\ln\pars{\verts{a}}}
\end{align}