Question

If a² + b² = 23ab, show that:

log `(a + b)/5 = 1/2`(log a + log b).

Answer 1

Given that

a² + b² = 23ab

⇒ a² + b² + 2ab = 23ab + 2ab

⇒ a² + b² + 2ab = 25ab

⇒ (a + b)² = 25ab

Taking log on both side

⇒ log(a + b)² = log25ab

⇒ 2log(a + b) = log25 + loga + logb

⇒ 2log(a + b) - log5² = loga + logb

⇒ 2log(a + b) - 2log5 = loga + logb

⇒ 2[log(a + b) - log5] = loga + logb

⇒ `log((a+b)/5)=1/2[loga+logb]`

Hence proved.