Question

If x² + y2 = 10xy, prove that 2 log (x + y) = log x + log y + 2log 2 + log 3.​

Answer 1

$\underline{\large{\sf To\:Prove :}}$

$\sf 2 log(x+y)= log x + log y + 2log 2 + log 3$

$\underline{\large{\sf Proof:}}$

$\sf x^2+y^2 = 10xy$....(Given)

Add 2xy on both the sides

∴ $\sf x^2 + y^2 +2xy = 10xy + 2xy$

{using identity $\sf (a+b)^2=a^2+2ab+b^2$ on LHS}

∴ $\sf (x + y)^2 = 12xy$

Now, taking log on both the sides

∴ $\sf log(x+y)^2= log (12xy)$

{using

$log_{e}( {m}^{n} ) = n log_{e}(m)$

on LHS }

∴ $\sf 2 log (x+y)=log (4\times3xy)$

{using

$log_{e}(mn) = log_{e}(m) + log_{e}(n)$

on RHS }

∴ $\sf 2 log (x+y)=log 4+ log 3+ log (xy)$

{we can write, log 4 = log (2)² = 2 log2 and log(xy)=log x + log y }

∴ $\sf 2 log(x+y)=2 log2 + log 3+log x + log y$

{rearrange the terms on RHS }

∴ $\boxed{\sf 2 log(x+y)= log x + log y + 2log 2 + log 3}$

Hence proved.