Question

prove that log75/16 - 2log5/9 + log32/243 = log2​

Answer 1

EXPLANATION.

Prove that.

⇒ ㏒(75/16) - 2㏒(5/9) + ㏒(32/243) = ㏒(2).

As we know that,

We can write equation as,

⇒ ㏒(75/16) - ㏒(5/9)² + ㏒(32/243).

⇒ ㏒(75/16) - ㏒(25/81) + ㏒(32/243).

⇒ ㏒(75/16/25/81) + ㏒(32/243).

⇒ ㏒(75/16 x 81/25) + ㏒(32/243).

⇒ ㏒(243/16) + ㏒(32/243).

⇒ ㏒[243/16 x 32/43].

⇒ ㏒(2).

Hence Proved.

                                                                                                                 

MORE INFORMATION.

Properties of logarithms.

Let M and N arbitrary positive number such that a > 0, a ≠ 1, b > 0, b ≠ 1 then.

(1) = ㏒ₐMN = ㏒ₐM + ㏒ₐN.

(2) = ㏒ₐ(M/N) = ㏒ₐM - ㏒ₐN.

(3) = ㏒ₐN^(α) = α㏒ₐN. (α any real number).

(4) = ㏒ₐ^(β)N^(α) = (α/β)㏒ₐN (α ≠ 0, β ≠ 0).

(5) = ㏒ₐN = ㏒_(b)N/㏒_(b)(a).

(6) = ㏒_(b) a. ㏒ₐb = 1 ⇒ ㏒_(b)a = 1/㏒ₐb.

(7) = e^(㏑a)ˣ = aˣ.