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

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

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

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

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

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

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

Combine like terms: lnx + -1lnx = 0
-4ln + 0 = ln(x + -1) + -1ln(2x + 1)
-4ln = ln(x + -1) + -1ln(2x + 1)

Reorder the terms:
-4ln = ln(-1 + x) + -1ln(2x + 1)
-4ln = (-1 * ln + x * ln) + -1ln(2x + 1)
-4ln = (-1ln + lnx) + -1ln(2x + 1)

Reorder the terms:
-4ln = -1ln + lnx + -1ln(1 + 2x)
-4ln = -1ln + lnx + (1 * -1ln + 2x * -1ln)
-4ln = -1ln + lnx + (-1ln + -2lnx)

Reorder the terms:
-4ln = -1ln + -1ln + lnx + -2lnx

Combine like terms: -1ln + -1ln = -2ln
-4ln = -2ln + lnx + -2lnx

Combine like terms: lnx + -2lnx = -1lnx
-4ln = -2ln + -1lnx

Solving
-4ln = -2ln + -1lnx

Solving for variable 'l'.

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

Add '2ln' to each side of the equation.
-4ln + 2ln = -2ln + 2ln + -1lnx

Combine like terms: -4ln + 2ln = -2ln
-2ln = -2ln + 2ln + -1lnx

Combine like terms: -2ln + 2ln = 0
-2ln = 0 + -1lnx
-2ln = -1lnx

Add 'lnx' to each side of the equation.
-2ln + lnx = -1lnx + lnx

Combine like terms: -1lnx + lnx = 0
-2ln + lnx = 0

Factor out the Greatest Common Factor (GCF), 'ln'.
ln(-2 + x) = 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 '(-2 + x)' equal to zero and attempt to solve:

Simplifying
-2 + x = 0

Solving
-2 + x = 0

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

Add '2' to each side of the equation.
-2 + 2 + x = 0 + 2

Combine like terms: -2 + 2 = 0
0 + x = 0 + 2
x = 0 + 2

Combine like terms: 0 + 2 = 2
x = 2

Add '-1x' to each side of the equation.
x + -1x = 2 + -1x

Combine like terms: x + -1x = 0
0 = 2 + -1x

Simplifying
0 = 2 + -1x

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.