How to compute Walrasian equilibrium in the case of logarithmic utility?

Question

If the utility is $U = \ln x + 2 \ln y$, how do you compute Walrasian equilibrium via usual formula for demand $x=a(x p_x + y p_y)/p_x(a+b)$ ?

What is $a$ and $b$?

In case of Cobb-Douglas function like $U=x^3y^4$ it would be simple: $a=3$ and $b=4$, but in this case how is it computed?

Answer 1

let $u \equiv xy^2$, then we have: $$U=lnx+2lny=ln(xy^2)=ln(u)$$ Since $U'(u)>0 \: \forall u>0$ it follows that the $(x,y)$ that maximizes $U$ also maximizes $u$; $\max \{U\}=\max\{ln(u)\}=ln(\max\{u\})$. $u$ represents the same preferences as $U$.

Clearly, $u(x,y)=x^a y^b$ with $a=1$, $b=2$.