Question

Log[a+b/2] = 1/2 [log a + log b] prove that a=b

Answer 1

log(a+b)/2=1/2(loga+logb)
or, log(a+b)/2=1/2 log(ab)
or, log(a+b)/2=log (ab)¹/²
or, (a+b)/2=√ab; (cancelling log from both sides)
or, (a+b)²/4=ab
or, (a²+2ab+b²)/4=ab
or, a²+2ab+b²=4ab
or, a²+2ab+b²-4ab=0
or, a²-2ab+b²=0
or, (a-b)²=0
or, a-b=0
or, a=b (proved)

Answer 2

$log( \frac{a+b}{2} ) = \frac{1}{2}(loga+logb)$
$log( \frac{a+b}{2} ) = \frac{1}{2}(logab)$
$log( \frac{a+b}{2} ) = log(ab)^\frac{1}{2}$
$\frac{a+b}{2} = \sqrt ab$

Squaring on both sides
$\frac{(a+b)^2}{4} = ab$

a² + b² + 2ab = 4ab

a² + b² - 2ab = 0 

(a - b)² = 0

a - b = 0

a = b