Question

If $a^x=N$ then ${\log }_{a}N=x$ and ${\log }_{2}ab={\log }_{2}a+{\log }_{2}b$. Find the number of solutions of the system of equation,

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

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (B)

$2$

We have, ${\log }_{2}xy=5\Rightarrow 2^5=xy..............\left[1\right]$

${\log }_{1/2}\left(x/y\right)=1\Rightarrow \frac{x}{y}=\frac{1}{2}\Rightarrow y=2x........\left[2\right]$

Substituting $2$ in $1$:

$2x^2=2^5\Rightarrow x^2=2^4\Rightarrow x=\pm 4$

For $x=4;y=8$ and

$x=-4;y=-8$.

Hence number of solutions are two, $(4,8)and(-4,-8)$