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Q. 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

Continuity and Differentiability

Solution:

$\log _2 x=4-\log _2 y \Rightarrow x=2^4 \cdot \frac{1}{2^{\log _2 y}}=\frac{16}{y}=x y=16$....(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
$t ^2-2 t -3=0 \Rightarrow( t -3)( t +1)=0$
$\log _y x=3 \Rightarrow x=y^3 $ and $\log _y x=-1 \Rightarrow x=1 / y$ ( no solution) ....(2)
from (1) and (2) $y^4=16 \Rightarrow y=2 \Rightarrow x=8$