Question No. 5. If log (xy³) = 1 and log (x²y) = 1, then log (xy) =
Answers
EXPLANATION.
⇒ ㏒(xy³) = 1. - - - - - (1).
⇒ ㏒(x²y) = 1. - - - - - (2).
As we know that,
We can evaluate equation one by one, we get.
From equation (1), we get.
⇒ ㏒(xy³) = 1. - - - - - (1).
As we know that,
Formula of :
⇒ ㏒ₐMN = ㏒ₐM + ㏒ₐN.
Using this formula in the equation, we get.
⇒ ㏒(x) + ㏒(y³) = 1.
⇒ ㏒(x) + 3㏒(y) = 1. - - - - - (3).
From equation (2), we get.
⇒ ㏒(x²y) = 1.
As we know that,
Formula of :
⇒ ㏒ₐMN = ㏒ₐM + ㏒ₐN.
Using this formula in the equation, we get.
⇒ ㏒(x²) + ㏒(y) = 1.
⇒ 2㏒(x) + ㏒(y) = 1. - - - - - (4).
From equation (3) and (4), we get.
⇒ ㏒(x) + 3㏒(y) = 1. - - - - - (3).
⇒ 2㏒(x) + ㏒(y) = 1. - - - - - (4).
Multiply equation (3) by 2.
Multiply equation (4) by 1.
⇒ ㏒(x) + 3㏒(y) = 1. - - - - - (3). x 2.
⇒ 2㏒(x) + ㏒(y) = 1. - - - - - (4). x 1.
We get,
⇒ 2㏒(x) + 6㏒(y) = 2. - - - - - (5).
⇒ 2㏒(x) + ㏒(y) = 1. - - - - - (6).
Subtract equation (5) and (6), we get.
⇒ 2㏒(x) + 6㏒(y) = 2. - - - - - (5).
⇒ 2㏒(x) + ㏒(y) = 1. - - - - - (6).
⇒ - - -
We get,
⇒ 5㏒(y) = 1.
⇒ ㏒(y) = 1/5.
Put the value of ㏒(y) = 1/5 in equation (3), we get.
⇒ ㏒(x) + 3㏒(y) = 1.
⇒ ㏒(x) + 3(1/5) = 1.
⇒ ㏒(x) = 1 - 3/5.
⇒ ㏒(x) = (5 - 3)/5.
⇒ ㏒(x) = 2/5.
To find :
⇒ ㏒(xy).
As we know that,
We can write equation as,
⇒ ㏒(xy) = ㏒(x) + ㏒(y).
⇒ ㏒(xy) = 2/5 + 1/5.
⇒ ㏒(xy) = 3/5.
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 and β ≠ 0).
(5) = ㏒ₐN = ㏒_{b}N/㏒_{b}a.
(6) = ㏒_{b} a . ㏒ₐ b = 1 ⇒ ㏒_{b}a = 1/㏒ₐb.
(7) = e^(㏑a)ˣ = aˣ.
✰REFER TO ATTACHMENT✰
THANKS!!
