There are 2 similar questions on $\log$ that I'm unable to solve.
Given that $\log_a xy^2 = p$ and $\log_a x^2/y^3 = q $. Express $\log_a 1/\sqrt{xy}$ or $\log_a 1/(xy)^{1/2}$ in terms of $p$ and $q$ ($a$ is the base). I was thinking along the line of using $p - q$ but I can't seem to get $y^{1/2}$. The answers are $3p+\frac{2q}7$ and $-5p-\frac q{14}$
Given that $\log_b(x^3y^2) = p$ and $\log_b(y/x) = q$. Express $\log_b(x^2y)$ in terms of $p$ and $q$ .($b$ is the base)
On
log(xy^2) = log(x) + 2 log(y) = p Eq-1 log(x^2/y^3) = 2 log(x) -3 log(y) = q Eq-2
solving Eq-1 and 2 by multiply Eq-1 by -2 both sides
-2 log(x) -4 log(y) = -2p 2 log(x) -3 log(y) = q ---------------------------- by adding -7 log(y) = q-2p log(y) = (2p - q)/7 now solve for log(x) the same way log(x) =(3p + 2q)/7
log(1/(xy)^1/2)) = 0 - 1/2(log(x)+log(y) = -1/2 log(x) -1/2 log(y0 we know log(x) and log(y) in term of p and q log(1/(xy)^1/2)) = -1/2(3p +2q)/7 -1/2(2p - q)/7 = -5p/14 -q/14 I hope that help
Hint: Try solving for $\log x$ and $\log y$ in terms of $p$ and $q$. Then use that result to get the values you're looking for.