ln(x-2)(x-2)=ln(x+1)+ln(x-1)

Solution for ln(x-2)(x-2)=ln(x+1)+ln(x-1) equation:

Simplifying
ln(x + -2)(x + -2) = ln(x + 1) + ln(x + -1)

Reorder the terms:
ln(-2 + x)(x + -2) = ln(x + 1) + ln(x + -1)

Reorder the terms:
ln(-2 + x)(-2 + x) = ln(x + 1) + ln(x + -1)

Multiply (-2 + x) * (-2 + x)
ln(-2(-2 + x) + x(-2 + x)) = ln(x + 1) + ln(x + -1)
ln((-2 * -2 + x * -2) + x(-2 + x)) = ln(x + 1) + ln(x + -1)
ln((4 + -2x) + x(-2 + x)) = ln(x + 1) + ln(x + -1)
ln(4 + -2x + (-2 * x + x * x)) = ln(x + 1) + ln(x + -1)
ln(4 + -2x + (-2x + x2)) = ln(x + 1) + ln(x + -1)

Combine like terms: -2x + -2x = -4x
ln(4 + -4x + x2) = ln(x + 1) + ln(x + -1)
(4 * ln + -4x * ln + x2 * ln) = ln(x + 1) + ln(x + -1)
(4ln + -4lnx + lnx2) = ln(x + 1) + ln(x + -1)

Reorder the terms:
4ln + -4lnx + lnx2 = ln(1 + x) + ln(x + -1)
4ln + -4lnx + lnx2 = (1 * ln + x * ln) + ln(x + -1)
4ln + -4lnx + lnx2 = (1ln + lnx) + ln(x + -1)

Reorder the terms:
4ln + -4lnx + lnx2 = 1ln + lnx + ln(-1 + x)
4ln + -4lnx + lnx2 = 1ln + lnx + (-1 * ln + x * ln)
4ln + -4lnx + lnx2 = 1ln + lnx + (-1ln + lnx)

Reorder the terms:
4ln + -4lnx + lnx2 = 1ln + -1ln + lnx + lnx

Combine like terms: 1ln + -1ln = 0
4ln + -4lnx + lnx2 = 0 + lnx + lnx
4ln + -4lnx + lnx2 = lnx + lnx

Combine like terms: lnx + lnx = 2lnx
4ln + -4lnx + lnx2 = 2lnx

Solving
4ln + -4lnx + lnx2 = 2lnx

Solving for variable 'l'.

Move all terms containing l to the left, all other terms to the right.

Add '-2lnx' to each side of the equation.
4ln + -4lnx + -2lnx + lnx2 = 2lnx + -2lnx

Combine like terms: -4lnx + -2lnx = -6lnx
4ln + -6lnx + lnx2 = 2lnx + -2lnx

Combine like terms: 2lnx + -2lnx = 0
4ln + -6lnx + lnx2 = 0

Factor out the Greatest Common Factor (GCF), 'ln'.
ln(4 + -6x + x2) = 0

Subproblem 1
Set the factor 'ln' equal to zero and attempt to solve:

Simplifying
ln = 0

Solving
ln = 0

Move all terms containing l to the left, all other terms to the right.

Simplifying
ln = 0

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Subproblem 2
Set the factor '(4 + -6x + x2)' equal to zero and attempt to solve:

Simplifying
4 + -6x + x2 = 0

Solving
4 + -6x + x2 = 0

Move all terms containing l to the left, all other terms to the right.

Add '-4' to each side of the equation.
4 + -6x + -4 + x2 = 0 + -4

Reorder the terms:
4 + -4 + -6x + x2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + -6x + x2 = 0 + -4
-6x + x2 = 0 + -4

Combine like terms: 0 + -4 = -4
-6x + x2 = -4

Add '6x' to each side of the equation.
-6x + 6x + x2 = -4 + 6x

Combine like terms: -6x + 6x = 0
0 + x2 = -4 + 6x
x2 = -4 + 6x

Add '-1x2' to each side of the equation.
x2 + -1x2 = -4 + 6x + -1x2

Combine like terms: x2 + -1x2 = 0
0 = -4 + 6x + -1x2

Simplifying
0 = -4 + 6x + -1x2

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.

The solution to this equation could not be determined.