Complex Analysis - Defining a branch cut for a function with multiple branches which DON'T lie on an axis

Question

For the function: $$f(z)=(z+\sqrt{3})^{1/2}\ln{(z-1)}$$ with the branch of this function chosen such that $$-\frac{4\pi}{3}<\arg{(z-1)}\leq\frac{2\pi}{3}$$ and $$-\frac{\pi}{2}<\arg{(z+\sqrt{3})}\leq\frac{3\pi}{2}$$ How would I find a single branch definition for this function? I am used to the method when both branch cuts lie on the $x$ or $y$ axis by analyzing the regions on the axis by showing if the function is the same above and below the axis and therefore whether a branch cut is required in this region. I am completely lost on what to do for this question since one of the branch cuts is not even parallel to an axis.