Question

. If x2 + y2= 25xy, then prove that 2 log(x + y) = 3log3 + logx +logy.
plz hlp to prove it​

Answer 1

$\large\underline\blue{\bold{Given \:  Question :-  }}$

$\bf \: If \: {x}^{2} + {y}^{2} = 25xy, \: prove \: that$

$\bf \:  2 log(x + y) = 3 log(3) + log(x) + log(y)$

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$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

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$\sf \:  ⟼ \: (1). \: log(x) + log(y) = log(xy)$

$\sf \:  ⟼ \: (2). \: log(x) - log(y) = log(\dfrac{x}{y} )$

$\sf \:  ⟼ \: (3). \: log( {x}^{y} ) = y log(x)$

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$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \large\underline\purple{\bold{Solution }}$

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$\sf \:  ⟼ \large \red {According \: to \: statement }$

$\sf \:  ⟼ \: {{x}^{2} + {y}^{2} = 25xy}$

☆ Adding 2xy on both side, we get

$\sf \:  ⟼ \: {x}^{2} + {y}^{2} + 2xy = 25xy + 2xy$

$\sf \:  ⟼ \: {(x + y)}^{2} = 27xy$

☆ Taking log on both sides, we get

$\sf \:  ⟼ \: log{(x + y)}^{2} = log(27xy)$

$\sf \:  ⟼ \: 2 log(x + y) = log(27) + log(x) + log(y)$

$\sf \:  ⟼ \: 2 log(x + y) = log( {3}^{3} ) + log(x) + log(y)$

$\sf \:  ⟼ \: 2 log(x + y) = 3log(3) + log(x) + log(y)$

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