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Since,
log (a+b)/2 = 1/2 log a + 1/2 log b
log (a+b)/2 = 1/2(log a + log b)
log (a+b)/2 = 1/2(log ab)
log (a+b)/2 = log (√ab)
⇒(a+b)/2 = √ab
Squaring on both sides we get
(a+b)²/4 = ab
(a+b)² = 4ab
a² + b² + 2ab - 4ab = 0
a² + b² - 2ab = 0
(a-b)² = 0
(a-b) = 0
⇒ a = b (proved)