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  4. The reals x and y satisfy log 8 x+ log 4(y2)=5 and log 8 y+ log 4(x2)=7 then the value of x y is

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Q. 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 $x y$ is

Continuity and Differentiability

Solution:

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