Utility functions and positive monotone transformations
Note that
$$ln(x_1)+2*ln(x_2) = ln(x_1)+ln(x_2^2) = ln (x_1 * x_2^2),$$
and note that
$$MRS_v = \frac{g'(x_1 * x_2^2) x_1 * 2 x_2}{g'(x_1 * x_2^2)   x_2^2}$$
such that the derivative of $g$ cancels out.
For more intuition, see here.