I understand how we define $e^z = e^x e^{iy}$ and use this to define the multi-valued function $\log(z) = \ln(r) + i(\theta + 2\pi n)$. I think I also understand how we may take any branch by setting an arbitrary real value, call it $\alpha$, and then restrict $\theta+2\pi n$ to the interval between $\alpha$ and $\alpha+2\pi$, so that the function becomes single-valued and still defined for all nonzero $z$.
What I'm particularly unclear on is what a branch cut is here. It is defined to be a curve used to describe a branch. (Brown and Churchill: "A branch cut is a portion of a line or curve that is introduced in order to define a branch F of a multiple-valued function $f$.") But where does this curve live? It doesn't seem to be in the domain or the range of the function.
For instance, any branch of $\log(z)$ is defined on the complex plane except the origin, so we cannot be restricting the domain. The image of any branch is the horizontal band with complex parts between $\alpha$ and $\alpha+2\pi$, and where the real part is strictly positive.
I might understand if the branch cut were some edge of this band. But later in Brown and Churchill they say that the branch cut is the origin and the ray $\theta=\alpha$. As far as I can tell, the points on this ray do not correspond to points in the domain or the range. For one thing, many different values of $\alpha$ pick out the same set of points, and yet every distinct value of $\alpha$ is (I think) supposed to correspond to a different branch.
Maybe that's ok because the definition of a branch cut did not specify where the cut must live, and just seems to use vague language about the cut being used to define a branch, but doesn't specify exactly how it must be used to accomplish this. But then that seems super open-ended and not usually how we define things in math.
I figure it's just more likely that I'm misunderstanding something.