Question

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

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{x^2+y^2=6\,xy}$

$\underline{\textbf{To prove:}}$

$\mathsf{2\;log(x+y)=log\,x+log\,y+3\,log\,2}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Product rule:}}$

$\boxed{\mathsf{log\,(MN)=log\,M+log\,N}}$

$\underline{\textbf{Power rule:}}$

$\boxed{\mathsf{log\,M^n=n\;log\,M}}$

$\mathsf{Consider,}$

$\mathsf{x^2+y^2=6\,xy}$

$\textsf{Add 2xy on bothsides, we get}$

$\mathsf{x^2+y^2+2\,xy=6\,xy+2\,xy}$

$\mathsf{(x+y)^2=8\,xy}$

$\textsf{Taking logrithm on bothsides, we get}$

$\mathsf{log\,(x+y)^2=log\,8\,xy}$

$\mathsf{2\,log\,(x+y)=log\,8+log\,x+log\,y}\;\;\;\;\textsf{(Using product and power rule)}$

$\mathsf{2\,log\,(x+y)=log\,2^3+log\,x+log\,y}$

$\boxed{\mathsf{2\,log\,(x+y)=3\,log\,2+log\,x+log\,y}}\;\;\;\;\textsf{(Using power rule)}$

Answer 2

Answer:

Given,

x²+y²=6xy

Need to Prove,

2log(x+y) = 3log2 + logx + logy