Question

Number of ordered pair(s) of (x,y) satisfying the system of equations, ${\log }_{2}xy=5$ and ${\log }_{\frac{1}{2}}\frac{x}{y}=1$ is:

• A. one
• B. two
• C. three
• D. four

Answer 1

The correct option is B two

Given,

${\log }_2\left(xy\right)=5,{\log }_{\frac{1}{2}}\left(\frac{x}{y}\right)=1$

${\log }_2\left(xy\right)=5$

$xy=2^5=32$

$x=\frac{32}{y}$

${\log }_{\frac{1}{2}}\left(\frac{x}{y}\right)=1$

${\log }_{\frac{1}{2}}\left(\frac{\frac{32}{y}}{y}\right)=1$

$\frac{\frac{32}{y}}{y}={\left(\frac{1}{2}\right)}^1$

$\frac{32}{y}=\frac{1}{2}y$

$y^2=64$

$\Rightarrow y=\pm 8$

$x=\frac{32}{\pm 8}=\pm 4$

$(4,8),(-4,-8)$ Therefore $2$ ordered pairs