$\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\isdiv}{\,\left.\right\vert\,} \newcommand{\ket}[1]{\left\vert #1\right\rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert} \newcommand{\wt}[1]{\widetilde{#1}}$ $\ds{a \in {\mathbb C}}$.
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}