Question

We have $a^x=N$, then ${\log }_{a}N=x$ and ${\log }_{2}ab={\log }_{2}a+{\log }_{2}b$. Then,

The number of solutions of the two given equations ${\log }_2\left(xy\right)=5$ and ${\log }_{0.5}\left(\frac{x}{y}\right)=1$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

${\log }_2\left(xy\right)=5$ and ${\log }_{0.5}\left(\frac{x}{y}\right)=1$

$\therefore xy=32$ and $\frac{x}{y}=\frac{1}{2}$

$\Rightarrow xy\times \frac{x}{y}=16\Rightarrow x^2=16\Rightarrow x=\pm 4$
Substituting the value of x in the equation $y=2x$, $y=\pm 8$
Thus (4, 8) and (-4, -8) are the possible solutions.

Hence the number of solutions $=2$.