Question

if X²+y²=7xy show that 2log(x+y)=logx+logy+2log3

Answer 1

(x+y)^2-2xy=7xy
(x+y)^2=9xy
log((x+y)^2)=log(((3)^2)xy)
2log(x+y)=logx+logy+2log3

Answer 2

Solution :-

x² + y² = 7xy

Adding 2xy on both sides

x² + y² + 2xy = 7xy + 2xy

(x + y)² = 9xy

[ Because (x + y)² = x² + y² + 2xy ]

(x + y)² = 3²xy

Taking log on both sides

$\sf \longrightarrow log(x + y)^2 = log 3^2xy \\ \\ \\ \sf \longrightarrow log(x + y)^2 = log 3^2 + logx + logy \\ \\ \\ \boxed{ \bf \because logab = loga + logb} \\ \\ \\ \sf \longrightarrow 2 log(x + y) = 2 log3 + logx + logy \\ \\ \\ \boxed{ \bf \because loga^m = m. loga } \\ \\ \\ \sf \longrightarrow 2 log(x + y) = logx + logy + 2log3$

Hence proved.

Laws of logarithms used :-

$\tt \longrightarrow logab = loga + logb$

$\tt \longrightarrow loga^m = m. loga$