Question

if log (a+b/4) =1/2 log ( ab), show that a² + b² = 14ab​

Answer 1

EXPLANATION.

To prove.

⇒ ㏒(a + b/4) = 1/2㏒(ab).

⇒ a² + b² = 14ab. [Given].

As we know that,

Formula of :

⇒ (a + b)² = a² + b² + 2ab.

⇒ (a + b)² = 14ab + 2ab.

⇒ (a + b)² = 16ab.

Taking log on both sides of the equation, we get.

⇒ ㏒[(a + b)²] = ㏒(16ab).

As we know that,

Formula of :

⇒ ㏒ₐ(MN) = ㏒ₐM + ㏒ₐN.

⇒ ㏒ₐα^(β) = β㏒ₐα.  (β > 0).

Using this formula in the equation, we get.

⇒ 2㏒(a + b) = ㏒(16) + ㏒(a) + ㏒(b).

⇒ 2㏒(a + b) = ㏒(4)² + ㏒(a) + ㏒(b).

⇒ 2㏒(a + b) = 2㏒(4) + ㏒(a) + ㏒(b).

⇒ 2㏒(a + b) - 2㏒(4) = ㏒(a) + ㏒(b).

As we know that,

Formula of :

⇒ ㏒ₐM/N. = ㏒ₐM - ㏒ₐN.

Using this formula in the equation, we get.

⇒ 2[㏒(a + b) - ㏒(4)] = ㏒(a) + ㏒(b).

⇒ 2[㏒(a + b)/4] = ㏒(a) + ㏒(b).

⇒ ㏒(a + b)/4 = 1/2[㏒(a) + ㏒(b)].

⇒ (a + b)/4 = 1/2 ㏒(ab).

Hence Proved.

Method = 2.

⇒ ㏒(a + b/4) = 1/2㏒(ab). [Given].

To prove.

⇒ a² + b² = 14ab.

As we know that,

We can write equation as,

⇒ 2㏒(a + b/4) = ㏒(ab).

⇒ ㏒(a + b/4)² = ㏒(ab).

⇒ (a + b/4)² = (ab).

⇒ [a² + b² + 2ab/16] = ab.

⇒ a² + b² + 2ab = 16ab.

⇒ a² + b² = 16ab - 2ab.

⇒ a² + b² = 14ab.

Hence Proved.

Answer 2

Answer:

To prove.

⇒ ㏒(a + b/4) = 1/2㏒(ab).

⇒ a² + b² = 14ab. [Given].

As we know that,

Formula of :

⇒ (a + b)² = a² + b² + 2ab.

⇒ (a + b)² = 14ab + 2ab.

⇒ (a + b)² = 16ab.

Taking log on both sides of the equation, we get.

⇒ ㏒[(a + b)²] = ㏒(16ab).

As we know that,

Formula of :

⇒ ㏒ₐ(MN) = ㏒ₐM + ㏒ₐN.

⇒ ㏒ₐα^(β) = β㏒ₐα.  (β > 0).

Using this formula in the equation, we get.

⇒ 2㏒(a + b) = ㏒(16) + ㏒(a) + ㏒(b).

⇒ 2㏒(a + b) = ㏒(4)² + ㏒(a) + ㏒(b).

⇒ 2㏒(a + b) = 2㏒(4) + ㏒(a) + ㏒(b).

⇒ 2㏒(a + b) - 2㏒(4) = ㏒(a) + ㏒(b).

As we know that,

Formula of :

⇒ ㏒ₐM/N. = ㏒ₐM - ㏒ₐN.

Using this formula in the equation, we get.

⇒ 2[㏒(a + b) - ㏒(4)] = ㏒(a) + ㏒(b).

⇒ 2[㏒(a + b)/4] = ㏒(a) + ㏒(b).

⇒ ㏒(a + b)/4 = 1/2[㏒(a) + ㏒(b)].

⇒ (a + b)/4 = 1/2 ㏒(ab).

Hence Proved.

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Method = 2.

⇒ ㏒(a + b/4) = 1/2㏒(ab). [Given].

To prove.

⇒ a² + b² = 14ab.

As we know that,

We can write equation as,

⇒ 2㏒(a + b/4) = ㏒(ab).

⇒ ㏒(a + b/4)² = ㏒(ab).

⇒ (a + b/4)² = (ab).

⇒ [a² + b² + 2ab/16] = ab.

⇒ a² + b² + 2ab = 16ab.

⇒ a² + b² = 16ab - 2ab.

⇒ a² + b² = 14ab.

Hence Proved.

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