Hello everyone!

Here are the problems I am struggling with:

1. Consider a consumer that derives utility from two goods, wine (W) and cloth (C), in the following way:
U(W,C)=logW +logC (1)
This consumer has income Y that has to spend in W and C. That is, this consumer has to decide optimally how many gallons of W to consume, and how many yards of C to consume. The market price of a gallon of W is pw, and the market price of one yard of cloth is pc. The consumer takes these market prices as given; he cannot influence them because he represents an insignificant part of the market.
(a) Write down the budget constraint faced by this consumer.
(b) Write down the consumerís problem (he maximizes 1 subject to the budget constraint).
(c) Write the first order conditions (FOC) of this problem (you will have two FOCs, one for W and one for C).
(d) Solve for the quantities consumed of W and C as a function of income Y , and prices pc and pw. [These are demand functions: equations that tell you how much of each good this consumer optimally wants to buy, given prices and income]

For this one I have gotten the budget constraint, Y = pw(W)+pc(C). For the maximization i have solved the Lagrangian to Lw = Uw(W,C) - Lambda(pw) = 0,
Lc = Uc(W,C) - Lambda(pc) = 0 and L lambda = Y - (pw(W)+pc(C))=0. I am very confused on what the First Order Conditions should be and how to proceed in solving the problem. I have another problem as well, but I am working on it and will post it later if i cannot solve it. Thank you!