Question

x2 + y2= 25xy, then prove that 2 log(x+y)=3log3+logx+logy

Answer 1

$\textbf{Formula used:}$

$\textbf{Product rule:}$

$\boxed{\bf\,log_a(MN)=log_aM+log_aN}$

$\textbf{Power rule:}$

$\boxed{\bf\,log_aM^n=n\;log_aM}$

$\textbf{Given:}$

$x^2+y^2=25xy$

$\textbf{To prove:}$

$2\log(x+y)=3\log3+\log\,x+\log\,y$

$\text{consider,}$

$(x+y)^2=x^2+y^2+2xy$

$(x+y)^2=25xy+2xy$

$(x+y)^2=27xy$

$\text{Taking logarithm on bothsides, we get}$

$\log(x+y)^2=\log(27xy)$

$\text{By using product rule}$

$\log(x+y)^2=\log3^3+\log\,x+\log\,y$

$\text{By using power rule}$

$\implies\,2\log(x+y)=3\log3+\log\,x+\log\,y$

$\therefore\boxed{\bf\,2\log(x+y)=3\log3+\log\,x+\log\,y}$

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Answer 2

Given :

x² + y² = 25xy

To Prove :

2 log(x+y) = 3log(3) + logx + logy

Proof :

x² + y² = 25xy (Given)

•Adding 2xy both sides

x² + y² + 2xy = 25xy + 2xy

•we Know that a² + b² + 2ab = (a+b)²

so,

(x+y)² = 27xy

•Taking log both sides

log (x+y)² = log ( 27xy)

•We know that

log(a)^n = n.log(a)

•Also,

log(a.b.c) = log a + log b + log c

These all are properties of log function

•therefore we get ,

2log(x+y) = log(27) + log(x) + log(y)

2log(x+y) = log(3)³ + log(x) + log(y)

2log(x+y) = 3log(3) + log(x) + log(y)

•Hence proved that ,

2log(x+y) = 3log(3) + log(x) + log(y)