Question

Solve : ${\log }_{2}xy=5$ , ${\log }_{\frac{1}{2}}\left(x/y\right)=1$

Answer 1

By definition, xy and $\frac{x}{y}$ are both + ive and
hence they can be written as $\left|xy\right|and\left|\frac{x}{y}\right|$
Thus we have the following equations.
$\left|xy\right|=\left|x\right|\left|y\right|=2^5$............(1)
$\left|\frac{x}{y}\right|=\frac{\left|x\right|}{\left|y\right|}=\frac{1}{2}$................(2)
Multiplying,${\left|x\right|}^2={\left|2\right|}^4=16$
$\therefore \left|x\right|=4$ only, as mod is +ive
$\therefore x=\pm 4y=\pm 8$
$\therefore x=4,y=8orx=-4y=-8only$
as both $xy$ and $\left(\frac{x}{y}\right)$ have to be + ive .