For the sake of simplicity, since we are always talking about base "b" let's just drop that symbol and use "log" to represent "log base b."

m = log(xy^3) = log(x) + log(y^3) = log(x) + 3*log(y)

p = log(x^3y^2) = log(x^3) + log(y^2) = 3*log(x) + 2*log(y)

Let's change the variables a bit and call

a = log(x)

b = log(y)

Then

m = a + 3b

p = 3a + 2b

Now solve for a and b by your favorite method and you get:

a = (3/7)p - (2/7)m = log(x)

b = -(1/7)p + 3/7)m = log(y)

We want the value of:

log(sqrt{xy}) = (1/2)*log(xy) = (1/2)*log(x) + (1/2)*log(y)

= (1/2)*[(3/7)p - (2/7)m] + (1/2)*[-(1/7)p + 3/7)m]

= (1/7)p + (1/14)m

-Dan