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Question
If a2 + b2 = 23ab, show that:
log `(a + b)/5 = 1/2`(log a + log b).
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Solution
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.
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