Question

If $\frac{\log x}{\log 5}=\frac{\log y^2}{\log 2}=\frac{\log 9}{\log \frac{1}{3}}$, find $x$ and $y$

Answer 1

$\frac{\log x}{\log 5}=\frac{\log 9}{\log \frac{1}{3}}$

$\Rightarrow \frac{\log x}{\log 5}=\frac{2\log 3}{-1\log 3}$

[$\log 9=\log 3^2=2\log 3,as\log a^m=m\log a,and,\log \frac{1}{3}=\log 3^{-1}=-1\log 3$]

$\Rightarrow \frac{\log x}{\log 5}=-2$

$\Rightarrow \log x=-2\log 5$

$\Rightarrow \log x=\log 5^{-2}$

$\therefore x=5^{-2}$

Now, $\frac{\log y^2}{\log 2}=\frac{\log 9}{\log \frac{1}{3}}$

$\Rightarrow \frac{\log y^2}{\log 2}=\frac{2\log 3}{-1\log 3}$

$\Rightarrow \frac{\log y^2}{\log 2}=-2$

$\Rightarrow \log y^2=-2\log 2$

$\Rightarrow \log y^2=\log 2^{-2}$ $[\because \log a^m=m\log a]$

$\Rightarrow y^2=2^{-2}$

$\Rightarrow y^2={\left(2^{-1}\right)}^2$

$\therefore y=\left(2^{-1}\right)$