Question

If $\log \left(x^2{y}^3\right)=a$, and $\log \frac{x}{y}=b$, find $\log x$ and $\log y$.

Answer 1

$\log \left(x^2{y}^3\right)=a--\left(1\right)\log \frac{x}{y}=b--\left(2\right)$

Eq (1) $\Rightarrow \log \left(x^2{y}^3\right)=a$

$\Rightarrow 2\log x+3\log y=a$

Eq (2) $\Rightarrow \log \left(\frac{x}{y}\right)=b$

$\Rightarrow \log x-\log y=b$

Solving both the equations gives

$\log x=\frac{1}{5}(a+3b),\log y=\frac{1}{5}(a-2b)$