Question

If a² + b² = 7ab, show that $2\log \Big(\frac{a+b}{3} \Big) = \log a + \log b$.

Answer 1

concepts : $log(AB)=logA+logB$.....(1)

$log\frac{A}{B}=logA-logB$........(2)

$logA^n=nlogA$.........(3)

it is given that, a² + b² = 7ab

or, a² + b² + 2ab = 2ab + 7ab

or, (a + b)² = 9ab

taking log both sides,

log(a + b)² = log(9ab)

or, 2log(a + b) = log(9ab) [ using formula (2), ]

or, 2log(a + b) = log9 + loga + logb [ using formula (1), ]

or, 2log(a + b) = log3² + loga + logb

or, 2log(a + b) - 2log3 = loga + logb

or, 2[ log(a + b) - log3 ] = loga + logb

or, 2log{ (a + b)/3 } = loga + logb

hence, it is clear that $2\log \Big(\frac{a+b}{3} \Big) = \log a + \log b$.