Question

iflog (a+b/2)=1/2 (loga+logb)then=​

Answer 1

Step-by-step explanation:

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Answer 2

QUESTION:

If log(a + b/2) = 1/2[log(a)+log(b)] then prove that a=b

USED FORMULAS:

log(a) + log(b) = log(ab)

2 log(a) = log(a)²

(a - b)² = a² + b² - 2ab

(a + b)² = a² + b² + 2ab

ANSWER:

Taking RHS

$\frac{1}{2} ( \log(ab))$

$= \log( {ab })^{ \frac{1}{2} }$

We know that power 1/2 equal square root

$= \log( \sqrt{ab} )$

Now,

$\log( \frac{ a + b}{2} ) = \log( \sqrt{ab} )$

Logarithms get cancelled

$\frac{a + b}{2} = \sqrt{ab} \\ \\ \frac{ {(a + b)}^{2} }{ {2}^{2} } = ab \\ \\ \frac{ {a}^{2} + {b}^{2} + 2ab }{4} = ab \\ \\ {a}^{2} + {b}^{2} = 4ab - 2ab \\ \\ {a}^{2} + {b}^{2} = 2ab \\ \\ {a}^{2} + {b}^{2} - 2ab = 0$

${(a - b)}^{2} = 0 \\ \\ a - b = 0 \\ \\ a = b$

Hence proved

CONCEPT USED:

⟩⟩ Logarithms