Logarithm URGENT?

Question

given that
p=log base a (x^3 *y^2) and q= log base a (y/x^5)
express log base a (x^1/3*y^1/3)

EDIT: express log base a (x^1/3*y^1/3) in terms of p and q

Answer 1

3 logx+ 2log y = p
log y - 5 log x = q
solving for log x & log y
log x = 1/13 ( p - 2q)
log y = 1/13 ( 5 p + 3 q)
So log (x^1/3*y^1/3) = 1/3 log x + 1/3 log y = 1/39(6p + q)

Answer 2

x^3 y^2 = a^p
y / x^5 = a^q

We want to write x^(1/3) y^(1/3) as a^(rp+sq) = a^(rp).a^(sq)for some r and s. So we need
(x^3 y^2)^r . (y / x^5)^s = x^(1/3) y^(1/3)
<=> x^(3r - 5s) . y^(2r + s) = x^(1/3) y^(1/3)
So we need
3r - 5s = 1/3
2r + s = 1/3
Solving these equations simultaneously gives us
13r = 2 => r = 2/13
13s = 1/3 => s = 1/39
So x^(1/3) y^(1/3) = a^(2p/13 + q/39)
and so log_a [x^(1/3) y^(1/3)] = 2p/13 + q/39.