Find a branch of $\log(3z-2)$ which is analytic at $z=0$ and takes the value $\ln 2+\pi i$ there. Specify the domain on which this branch is analytic.
These are my thoughts. I know that $\log z$ is multivalued function and $\log(3z-2)=\log|3z-2|+i(\text{Arg} z+2\pi n)$, where $n\in \mathbb{Z}$.
Can anyone show the solution, please?
For each $z\in\Bbb C\setminus[0,\infty)$, let $\operatorname{Log}(z)=\log|z|+\operatorname{arg}(z)i$, where $\operatorname{arg}(z)$ is the only argument of $z$ such that $\operatorname{arg}(z)\in(0,2\pi)$. Then $\operatorname{Log}$ is analytic. Now, let $l(z)=\operatorname{Log}(3z-2)$. Then $l$ is a branch of $\log(3z-2)$ and $l(0)=\operatorname{Log}(-2)=\log2+\pi i$.