Question

The reals $x$ and y satisfy ${\log }_{8}x+{\log }_4\left(y^2\right)=5$ and ${\log }_{8}y+{\log }_4\left(x^2\right)=7$ then the value of $xy$ is

• A. 1024
• B. 512
• C. 256
• D. 81

Answer 1

Answer: (B)

$\text{ equation(1) }\Rightarrow \frac{1}{3}{\log }_{2}x+{\log }_{2}y=\mathrm{5......}(3)$ and
$\text{ equation(2) }\Rightarrow \frac{1}{3}{\log }_{2}y+{\log }_{2}x=7$ .....(4)
$(3)+(4)\Rightarrow \frac{1}{3}{\log }_2(xy)+{\log }_2(xy)=12$
$\Rightarrow \frac{4}{3}{\log }_2(xy)=12$
${\log }_2(xy)=9\Rightarrow xy=2^9=512$