a branch cut

A multi-valued function becomes usable only after we forbid the loops that change its value. A branch cut is the curtain we draw to do exactly that: a curve we delete from the plane so that no path inside the remaining region can encircle a branch point, which lets a single-valued branch live there continuously.

For the logarithm the standard cut is the negative real axis, from 0 out to minus infinity, used by the principal branch Log z. Once that ray is removed, the slit plane has no loop that circles the origin, so Arg z can be chosen continuously in (-pi, pi] and Log z = ln|z| + i Arg z is a genuine single-valued holomorphic function on the cut plane. The cut is not unique: any curve from the branch point 0 to infinity will do, for instance the negative imaginary axis or a spiral. You choose the cut to keep the region you care about (your domain of interest) intact and away from the seam.

The price of the cut is a controlled discontinuity along it: the function takes different limits on the two sides of the curve (for Log z the jump is exactly 2 pi i across the negative real axis). That is acceptable because you simply never let your contours or computations cross the cut. For a function with two finite branch points, like sqrt((z - 1)(z + 1)), a smart cut can be the finite segment joining them, after which loops that enclose both branch points together cause no contradiction and the function is single-valued outside the segment. Choosing where to cut is an art guided by where the branch points are and what region must stay connected and uncut.