I am trying to construct a nonempty open subset $D$ of the annulus $\{z:1<|z|<2\}$ such that (i) $D$ is connected and so is its boundary, (ii) a holomorphic branch of $\log z$ can be defined on $D$, (iii) $|\log z|$ is unbounded on $D$.
I was thinking that since $\ln|z|$ will be bounded on $D$, I will need $\arg z$ to be unbounded on $D$ to get (iii). To get (ii), I think I just need to be able to define $\arg z$ continuously on $D$. What came to my mind was to have a boundary curve that somehow spirals endlessly inside the annulus but I don't know how to make it precise and also to consider condition (i).
Update: I have arrived at the spiral curve given by $r=1+\frac{1}{\theta-1}$ for $\theta\geq\pi$. This curve is contained in the annulus and spirals around the circle $|z|=1$, getting closer to the circle as it does so. I can take a suitable neighborhood of the curve as $D$. Now what I am missing is the justification that the boundary of $D$ will be connected.