Question

If $\log \frac{2x}{y}=1$ and $\frac{2}{3}x=\frac{5}{4}{y}^2$, then find the value of $x$ and $y.$

Answer 1

Since, $\log \frac{2x}{y}=1$
$\log \frac{2x}{y}=\log 10$
$\Rightarrow 2x/y=10$ ..... (1)
But $\frac{2}{3}x=\frac{5}{4}{y}^2$
$\therefore x=\frac{3}{2}\times \frac{5}{4}.y^2=\frac{15}{8}{y}^2$
Now, from (1)
$\frac{2\times \frac{15}{8}.y^2}{y}=10$
$\therefore y=\frac{10\times 4}{15}=\frac{8}{3}=2\frac{2}{3}$
$\therefore x=\frac{15}{8}\times {\left(\frac{8}{3}\right)}^2=\frac{15}{8}\times \frac{8^2}{3^2}$
$=\frac{120}{9}=\frac{40}{3}=13\frac{1}{3}$
$\Rightarrow x=13\frac{1}{3}$
$y=2\frac{2}{3}$