Microeconomics,Trade/Exchange problem

[alias]

There are 2 consumers, A and B with the following utility functions and endowments:
uA(xA1, xA2) = a ln xA1 + (1-a) ln xA2 , endowment for A is wA = (0, 1)

uB(xB1,xB2) = min(xB1,xB2) endowment for B is wB = (1,0)

Find the market clearing prices and the equilibrium allocation

From Solutions Manual:

x1A = ay=p1 = ap2=p1; x1B = x2B
so from budget constraint, (p1 +p2)x1B = p1,
so x1B =p1=(p1+p2). Choose p1 = 1 an numeraire and solve
ap2 +1/(1 + p2) = 1.

My textbook is not helpful and I don't know how to get to the answer in the solutions manual. Can anyone give me some help? I'm very lost.
Thanks

 

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! The market clearing prices can be found by setting up the following system of equations:Consumer A's budget constraint: p1xA1 + p2xA2 = wA p1xA1 + p2(1) = 0 => p1xA1 = -p2 Consumer B's budget constraint: p1xB1 + p2xB2 = wB p1(1) + p2xB2 = 1 => p2xB2 = 1 - p1 Equilibrium allocation: xA1 = ay = p1 xA2 = (1-a)y = (1-a)p2 xB1 = xB2 = y = p1 Solving for the market clearing prices, we have: ap2 + 1/(1+p2) = 1 => p2 = (1-a)/(a+1) and p1 = 1 Therefore, the market clearing prices are p1 = 1 and p2 = (1-a)/(a+1). The equilibrium allocation is xA1 = ap1 = a, xA2 = (1-a)p2 = (1-a)(1-a)/(a+1), xB1 = xB2 = y = p1 = 1.