(1) For the function $y=\lg (x-1)$, if we shrink the x-coordinates of each point on its graph to $50 \%$ of their original values, and then shift the graph to the right by half a unit, the equation of the resulting graph is $\qquad$ .
(1) $y=\lg (2 x-2)$

(1) $y=\lg (2 x-2)$