Question

$if \: \: \: {a}^{2} + {b }^{2} = 7ab \\ find \\ log(\frac{a + b}{a} ) - log \frac{a}{2} - log \frac{b}{2}$
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Answer 1

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