Question

if 2log(x-y)=logx +logy, prove that 2log(x+y)= log5+logx +logy

Answer 1

$\textbf{Given:}$

$2\,log(x-y)=log\,x+log\,y$

$\textbf{To find:}$

$2\,log(x+y)=log\,5+log\,x+log\,y$

$\textbf{Solution:}$

$\textbf{Formula used:}$

$\textbf{Product rule:}$

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

$\textbf{Power rule:}$

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

$\text{Consider,}$

$2\,log(x-y)=log\,x+log\,y$

$\text{Using power and product rule, we get}$

$log(x-y)^2=log\,xy$

$\implies\,(x-y)^2=xy$

$\text{Adding 4xy on both sides we get}$

$(x-y)^2+4xy=xy+4xy$

$x^2+y^2-2xy+4xy=5xy$

$x^2+y^2+2xy=5xy$

$(x+y)^2=5xy$

$\text{Taking logarithm on both sides, we get}$

$log(x+y)^2=log(5xy)$

$\text{Using power rule on left side and product rule on right side, we get}$

$2\,log(x+y)=log\,5+log\,xy$

$\implies\bf\,2\,log(x+y)=log\,5+log\,x+log\,y$

Find more:

x62 + y^2= 25xy, then prove that 2 log(x+y)=3log3+logx+logy

https://brainly.in/question/477579