Attach analytic branch of some complex logarithm function and find range (image) of set if possible
radius $\arg z =\pi /6$
positive $y$ axis
circle $|z|=1$
ring $3 \leq |z| \leq5$
I'm not sure how to start this but, $\log z=\ln r + i\theta$. And $\operatorname{Im} (\log z)=\theta$. Do I need to find $\log z$ for which $\operatorname{Im} (\log z)=\pi/6$ or i'm I completely lost.
radius $\arg z =\pi /6$ $\to$ $z=re^{i\pi/6}$
$$ \log z = \ln r + i \pi /6$$
Positive $y$ axis $\to$ $z=re^{i\pi/2}$ $$\log z=\ln r + i\pi /2$$
circle $|z|=1$ $\to$ $z=e^{i\theta}$ $$\log z=\ln 1+i\theta$$
ring $3 \leq |z| \leq5$ $\to$ $$\log z=\ln r + i\theta, \; \text{where} 3\leq r \leq 5$$ Did I get these correctly? critique?