The region of analyticity of log(sqrt(z)). (using principle branch)

I know the domain of log(z) is the complex plane minus the origin and the negative x-axis (C\{x+iy | x =< 0, y=0}). I'm not sure what domain of sqrt(z) produces this as an image, though. I know z --> sqrt(z) cuts angles in half... sooo would the domain be everything but the positive x-axis? I'm a little confused