I really don't understand this:
The set of values of is , with , and that the same is true of , but...
The set of values of is not the same as the set of values of .
I think that the point here is that the square root function, like the logarithm, is multivalued (or 2-valued in the case of the square root). So the set of values of means the set of all the logarithms of both square roots of z.
But when you multiply by 2, you only get half the values of (the other half come from ).