Transformation of Cobb Douglas Utility So, I need to determine whether or not $u=x^{0.5}y^{0.5}$ exhibits the same preferences as $u'= log(x) + log (y)$. Any tricks I can use here? I've done the logarithmic transformation so that I now have $(0.5)log (x) + (0.5)log (y)$. My guess is, if I multiply these by $2$ I'll get $log (x) + log (y)$ and thereby proving that u and u' refer to the same preferences, but I'm not sure if this is correct?
Hint: how does a positive monotone transformation of the utility function affect the consumer's maximization problem?

Edit,

Suppose the optimal bundle for some agent with some utility function $u$ is given by $x^{*} = (x_{1}^{*},x_{2}^{*})$. Suppose then that I take the transformation of $u$ by some strictly monotone function $f$. I claim that $x^{*} = (x_{1}^{*},x_{2}^{*})$ is the optimal bundle for the new utility function $v = f \circ u$. 

Suppose not, suppose that there is another bundle $\hat{x} = (\hat{x}_{1},\hat{x}_{2})$ such that $v(\hat{x}) &gt; v(x^{*})$. But by the definition of $v$, we have $$\tag{In}\label{In} f(u(\hat{x})) &gt; f(u(x^{*}))$$ Then, since $f$ is strictly monotone, it is invertible and so from \ref{In} we must have $u(\hat{x}) &gt; u(x^{*})$, which contradicts optimality of $x^{*}$.

The $\ln$ function is one such strictly monotone function on domain $\Re_{+}$.