Utility functions and positive monotone transformations We let $g(z)$ be a strictly monotonous function so:
$$\frac{dg(z)}{dz}&gt;0$$
Consumer 1 has preferences given by the utility function $u(x_1,x_2)=ln(x_1)+2ln(x_2)$, while consumer 2 has preferences given by n $v(x_1,x_2)=g(x_1x_2^2)$.
Then I have to show that consumer 2 got same preferences as consumer 1.
I think I have to use MSR on $x_1x_2^2$ and on $v(x_1,x_2)$. For MSR on $x_1x_2^2$  I get:
$$MRS=-\frac{\frac{\partial }{\partial x_1}}{\frac{\partial }{\partial x_2}}=-\frac{x_2^2}{2x_1x_2}$$
But How can I find MSR on $v(x_1,x_2)$ (I think I got the same MRS if I use the chain rule, but I'm not sure?), and how can I use this to conclude that the ranking of the two indifference curves is the same when $g(z)$ is monotonous?
I hope that someone can help me?
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.