Question

x2+y2=6xy, prove that 2 log (x+y) =logx+logy+3log2​

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:2\:\log\:(\:x\:+\:y\:)\:=\:\log\:(\:x\:)\:+\:\log\:(\:y\:)\:+\:3\:\log\:(\:2\:)}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:x^2\:+\:y^2\:=\:6xy}$

We have to prove that,

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

Now,

$\displaystyle{\sf\:x^2\:+\:y^2\:=\:6xy}$

$\displaystyle{\implies\sf\:(\:x\:+\:y\:)^2\:-\:2xy\:=\:6xy}$

$\displaystyle{\implies\sf\:(\:x\:+\:y\:)^2\:=\:6xy\:+\:2xy}$

$\displaystyle{\implies\sf\:(\:x\:+\:y\:)^2\:=\:8xy}$

By taking log on both sides, we get,

$\displaystyle{\implies\sf\:\log\:[\:(\:x\:+\:y\:)^2\:]\:=\:\log\:(\:8xy\:)}$

We know that,

$\displaystyle{\bullet\:\pink{\sf\:\log_b\:(\:a^k\:)\:=\:k\:\log_b\:(\:a\:)}}$

$\displaystyle{\bullet\:\pink{\sf\:\log_b\:(\:mn\:)\:=\:\log_b\:(\:m\:)\:+\:\log_b\:(\:n\:)}}$

$\displaystyle{\implies\sf\:2\:\log\:(\:x\:+\:y\:)\:=\:\log\:(\:8\:)\:+\:\log\:(\:x\:)\:+\:\log\:(\:y\:)}$

$\displaystyle{\implies\sf\:2\:\log\:(\:x\:+\:y\:)\:=\:\log\:(\:x\:)\:+\:\log\:(\:y\:)\:+\:\log\:(\:8\:)}$

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

$\displaystyle{\implies\:\underline{\boxed{\red{\sf\:2\:\log\:(\:x\:+\:y\:)\:=\:\log\:(\:x\:)\:+\:\log\:(\:y\:)\:+\:3\:\log\:(\:2\:)}}}}$

Hence proved!

Answer 2

Given :-

$\:$

$\bf⟼ \: \: {x}^{2} + {y}^{2} \: = \: 6xy$

$\:$

To Prove :-

$\:$

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

$\:$

Solution :-

$\:$

Given that,

$\:$

$\bf⟼ \: \: {x}^{2} + {y}^{2} \: = \: 6xy$

$\:$

Adding 2xy on both sides,

$\:$

$\bf⟼ \: \: {x}^{2} + {y}^{2} +2xy\: = \: 6 xy+ 2xy$

$\:$

$\bf⟼ \: \: ({x} + {y})^{2} \: = \: 8xy \: \: \: \: \: \: \: \: \: \bigg( ∴( {a + b)}^{2} \: = \: {a}^{2} + {b}^{2} + 2ab \bigg)$

$\:$

Adding log on both sides,

$\:$

$\bf⟼ \: \: log \: ({x} + {y})^{2} \: = \: log \: 8xy$

$\:$

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

$\:$

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

$\:$

Hence, proved !!!

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