Question

if 3log(x^2y)=4+2logx-logy where x,y>0; express y interms x. if x-y=2√6, find the value of x and y​

Answer 1

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$\underline{\blacksquare\:\:\:\footnotesize{\red{\text{SolutioN}}}}$

$\footnotesize{\therefore\:\:3log(x^2y)=4+2log(x)-log(y)}$

$\footnotesize{\implies\:\:3\big(log(x^2)+log (y)\big)=4+2log(x)-log(y)}$

$\footnotesize{\implies\:\:3\big(2log(x)+log (y)\big)=4+2log(x)-log(y)}$

$\footnotesize{\implies\:\:6log(x)+3log (y)\big)=4+2log(x)-log(y)}$

$\footnotesize{\implies\:\:6log(x)+3log (y)-2log(x)+log(y)=4}$

$\footnotesize{\implies\:\:4log(x)+4log (y)=4}$

$\footnotesize{\implies\:\:4\big(log(x)+log (y)\big)=4}$

$\footnotesize{\implies\:\:\big(log(x)+log (y)\big)=\dfrac{\cancel4}{\cancel4}}$

$\footnotesize{\implies\:\:\big(log(x)+log (y)\big)=1}$

$\footnotesize{\implies\:\:log(xy)=log10}$

$\footnotesize{\implies\:\:\red{xy=10}}$

$\footnotesize{\implies\:\:\red{\boxed{y=\dfrac{10}{x}}}}$

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$\footnotesize{\text{Now given that , }}$

$\footnotesize{\therefore\:\:x-y=2\sqrt{8}}$

$\footnotesize{\implies\:\:x-\dfrac{10}{x}=2\sqrt{8}}$

$\footnotesize{\implies\:\:\dfrac{x^2-10}{x}=4\sqrt{2}}$

$\footnotesize{\implies\:\:x^2-10=4\sqrt{2}\:x}$

$\footnotesize{\implies\:\:x^2-4\sqrt{2}\:x-10=0}$

$\footnotesize{\implies\:\:x^2-5\sqrt{2}\:x+\sqrt{2}\:x-10=0}$

$\footnotesize{\implies\:\:x(x-5\sqrt{2})+\sqrt{2}(x-5\sqrt{2})=0}$

$\footnotesize{\implies\:\:(x-5\sqrt{2})(x+\sqrt{2})=0}$

$\footnotesize{\text{either ,}\:\:\:(x-5\sqrt{2})=0}$

$\footnotesize{\:\:\:\implies\:\red{\boxed{x=5\sqrt{2}}}}$

$\footnotesize{\text{or ,}\:\:\:(x+\sqrt{2})=0}$

$\footnotesize{\:\:\:\implies\:\red{\boxed{x=-\sqrt{2}}}}$

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$\footnotesize{\red{\text{when ,}\:\:\:x=5\sqrt{2}}}$

$\footnotesize{\therefore\:\:x-y=4\sqrt{2}}$

$\footnotesize{\implies\:\:y=5\sqrt{2}-4\sqrt{2}}$

$\footnotesize{\implies\:\:\red{y=\sqrt{2}}}$

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$\footnotesize{\red{\text{when ,}\:\:\:x=-\sqrt{2}}}$

$\footnotesize{\therefore\:\:x-y=4\sqrt{2}}$

$\footnotesize{\implies\:\:y=-\sqrt{2}-4\sqrt{2}}$

$\footnotesize{\implies\:\:\red{y=-5\sqrt{2}}}$

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