lnx=ln(x-1)+1.Solve for x.
Answers
EXPLANATION.
⇒ ㏑(x) = ㏑(x - 1) + 1.
As we know that,
We can write equation as,
⇒ ㏑(x) - ㏑(x + 1) = 1.
As we know that,
Formula of :
⇒ ㏑(a) - ㏑(b) = ㏑(a/b).
⇒ ㏑(e) = 1.
Using this formula in the equation, we get.
⇒ ㏑[(x)/(x - 1)] = 1.
⇒ ㏑[(x)/(x - 1)] = ㏑(e).
⇒ (x)/(x - 1) = e.
⇒ x = e(x - 1).
⇒ x = ex - e.
⇒ x - ex = - e.
⇒ x(1 - e) = - e.
⇒ x = (-e)/(1 - e).
⇒ x = (e)/(e - 1).
MORE INFORMATION.
Properties of logarithms.
Let M & 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 numbers).
(4) = ㏒_(α)^(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.
EXPLANATION.
⇒ ㏑(x) = ㏑(x - 1) + 1.
As we know that,
We can write equation as,
⇒ ㏑(x) - ㏑(x + 1) = 1.
As we know that,
Formula of :
⇒ ㏑(a) - ㏑(b) = ㏑(a/b).
⇒ ㏑(e) = 1.
Using this formula in the equation, we get.
⇒ ㏑[(x)/(x - 1)] = 1.
⇒ ㏑[(x)/(x - 1)] = ㏑(e).
⇒ (x)/(x - 1) = e.
⇒ x = e(x - 1).
⇒ x = ex - e.
⇒ x - ex = - e.
⇒ x(1 - e) = - e.
⇒ x = (-e)/(1 - e).
⇒ x = (e)/(e - 1).