Someone please help me with this Math!?

Question

Find x:

ln(x+5) = ln(x-1) - ln(x+1)

i have no idea where to start! i knew at one point but its been a year or so and i forgot

Answer 1

1. Use property ln(a) - ln(b) = ln(a/b):
ln(x+5) = ln((x-1)/(x+1))

2. Exponentiate both sides:
x + 5 = (x-1)/(x+1)

3. Multiply through by (x+ 1):
x^2 + 6x + 5 = x-1

4. Collect terms and write in standard form:
x^2 + 5 x + 6 = 0

5. Factor into (x+2)(x+3) = 0, and get your answer!
x = -2, -3

Answer 2

We use two properties of logarithms: ln A - ln B = ln (A/B) and ln C = ln D if and only if C = D. Now,
ln(x + 5) = ln(x - 1) - ln(x + 1) = ln((x - 1)/(x + 1). Then
x + 5 = (x - 1)/(x + 1) or (x + 5)*(x + 1) = x - 1. This says that
x^2 + 6x + 1 = x - 1 , so x^2 + 5x + 6 =0. But then
(x + 3)*(x + 2) = 0, so x = -3 or x = -2.

Now the right side of your given equation does not make sense for these values, since logs of negative numbers are not defined.

Answer 3

Tony is perfectly right, but begin with this:
x + 5 > 0, x - 1 > 0, x + 1 > 0, solving them you obtain that x must satisfy the following: x > 1! Then proceed as shown in the both answers above, and finally, when both values found DO NOT satisfy x > 1, come to the conclusion: the equation has no solutions.