Question

Number of ordered pair(s) (x,y) simultaneously satisfying the system of equations ${\log }_{y}x-3{\log }_{x}y=2$ and ${\log }_{2}x=4-{\log }_{2}y$ is/are

• A. 1
• B. 2
• C. 3
• D. none

Answer 1

Answer: (A)

${\log }_{2}x=4-{\log }_{2}y\Rightarrow x=2^4\cdot \frac{1}{2^{{\log }_{2}y}}=\frac{16}{y}=xy=16$....(1)
and ${\log }_{y}x-3{\log }_{x}y=2\Rightarrow {\left({\log }_{y}x\right)}^2-2{\log }_{y}x-3=0$
let ${\log }_{y}x=t$ the equation becomes
$t^2-2t-3=0\Rightarrow (t-3)(t+1)=0$
${\log }_{y}x=3\Rightarrow x=y^3$ and ${\log }_{y}x=-1\Rightarrow x=1/y$ ( no solution) ....(2)
from (1) and (2) $y^4=16\Rightarrow y=2\Rightarrow x=8$