Question

if x²+y²=105xy then prove 2log(x-y)=log3+log5+log7+logx+logy​

Answer 1

Given :

The given equation is

•x²+y²=105xy

To Prove :

2log(x-y)=log3+log5+log7+logx+logy

Formula :

$\bullet \bf \: log(xy) = logx + logy$

$\bullet \bf \: log \bigg( \frac{x}{y} \bigg) = logx - logy$

Solution :

$\bullet \tt \: the \: given \: equation \: is :$

$\to \bf {x}^{2} + {y}^{2} = 107xy$

Subtracting 2xy on both sides :

$\\ \implies \sf \: {x}^{2} + {y}^{2} - 2xy = 107xy - 2xy$

$\implies \sf \: {(x - y)}^{2} = 105xy$

Getting log both sides :

$\\ \implies \sf \: {log}^{ {(x - y)}^{2} } = {log}^{105xy}$

$\implies \sf \: 2 {log}^{(x - y)} = {log}^{105xy}$

$\\ \implies \sf \: 2 {log}^{(x - y)} = {log}^{(3 \times 5 \times 7 \times x \times y)}$

$\\ \implies \sf \: {2log}^{(x - y)} = {log}^{3} + {log}^{5} + {log}^{7} + {log}^{x} + {log}^{y}$

$\\ \implies \boxed{ \bf{ {2log}^{(x - y)} = {log}^{3} + {log}^{5} + {log}^{7} + {log}^{x} + {log}^{y} }}$

Answer 2

Answer:

Step-by-step explanation: