Logarithmic equations in one variable

Question

If $\log x\cdot y^3 =m$ and $\log x^3\cdot y^2=p$, then find $\log (\frac{x^2}{y})$.

Please show all the steps necessary to solve the above.

Thanks in advance.

Answer 1

Hint: Recall that if $b, u, v > 0$, with $b \neq 1$, then $\log_b(\frac{u}{v}) = \log_b u - \log_b v$. Then $\log x^3y^2 - \log xy^3 = $

Answer 2

To solve this you need to remember the properties of logarithms. $$ \log(ab) = \log(a) +\log(b) $$

And

$$ \log(a^b) = b \cdot \log(a) $$

Using these two rules you can rewrite "m" and "p" as:

$$ m= \log(x)+3 \cdot \log(y) $$ $$ p=3 \cdot \log(x)+2 \cdot \log(y) $$

From here you can rearrange and solve not for "x" and "y" but for "log(x)" and "log(y)". Remember you are looking for $ \log(x^2/y) = 2 \cdot \log(x) - \log(y)$