2ln(x+1)+ln(3x+5)+ln(2)=ln(6x+1)+ln(x-2)+ln(x-2)

Solution for 2ln(x+1)+ln(3x+5)+ln(2)=ln(6x+1)+ln(x-2)+ln(x-2) equation:

Simplifying
2ln(x + 1) + ln(3x + 5) + ln(2) = ln(6x + 1) + ln(x + -2) + ln(x + -2)

Reorder the terms:
2ln(1 + x) + ln(3x + 5) + ln(2) = ln(6x + 1) + ln(x + -2) + ln(x + -2)
(1 * 2ln + x * 2ln) + ln(3x + 5) + ln(2) = ln(6x + 1) + ln(x + -2) + ln(x + -2)
(2ln + 2lnx) + ln(3x + 5) + ln(2) = ln(6x + 1) + ln(x + -2) + ln(x + -2)

Reorder the terms:
2ln + 2lnx + ln(5 + 3x) + ln(2) = ln(6x + 1) + ln(x + -2) + ln(x + -2)
2ln + 2lnx + (5 * ln + 3x * ln) + ln(2) = ln(6x + 1) + ln(x + -2) + ln(x + -2)
2ln + 2lnx + (5ln + 3lnx) + ln(2) = ln(6x + 1) + ln(x + -2) + ln(x + -2)

Reorder the terms for easier multiplication:
2ln + 2lnx + 5ln + 3lnx + 2ln = ln(6x + 1) + ln(x + -2) + ln(x + -2)

Reorder the terms:
2ln + 5ln + 2ln + 2lnx + 3lnx = ln(6x + 1) + ln(x + -2) + ln(x + -2)

Combine like terms: 2ln + 5ln = 7ln
7ln + 2ln + 2lnx + 3lnx = ln(6x + 1) + ln(x + -2) + ln(x + -2)

Combine like terms: 7ln + 2ln = 9ln
9ln + 2lnx + 3lnx = ln(6x + 1) + ln(x + -2) + ln(x + -2)

Combine like terms: 2lnx + 3lnx = 5lnx
9ln + 5lnx = ln(6x + 1) + ln(x + -2) + ln(x + -2)

Reorder the terms:
9ln + 5lnx = ln(1 + 6x) + ln(x + -2) + ln(x + -2)
9ln + 5lnx = (1 * ln + 6x * ln) + ln(x + -2) + ln(x + -2)
9ln + 5lnx = (1ln + 6lnx) + ln(x + -2) + ln(x + -2)

Reorder the terms:
9ln + 5lnx = 1ln + 6lnx + ln(-2 + x) + ln(x + -2)
9ln + 5lnx = 1ln + 6lnx + (-2 * ln + x * ln) + ln(x + -2)
9ln + 5lnx = 1ln + 6lnx + (-2ln + lnx) + ln(x + -2)

Reorder the terms:
9ln + 5lnx = 1ln + 6lnx + -2ln + lnx + ln(-2 + x)
9ln + 5lnx = 1ln + 6lnx + -2ln + lnx + (-2 * ln + x * ln)
9ln + 5lnx = 1ln + 6lnx + -2ln + lnx + (-2ln + lnx)

Reorder the terms:
9ln + 5lnx = 1ln + -2ln + -2ln + 6lnx + lnx + lnx

Combine like terms: 1ln + -2ln = -1ln
9ln + 5lnx = -1ln + -2ln + 6lnx + lnx + lnx

Combine like terms: -1ln + -2ln = -3ln
9ln + 5lnx = -3ln + 6lnx + lnx + lnx

Combine like terms: 6lnx + lnx = 7lnx
9ln + 5lnx = -3ln + 7lnx + lnx

Combine like terms: 7lnx + lnx = 8lnx
9ln + 5lnx = -3ln + 8lnx

Solving
9ln + 5lnx = -3ln + 8lnx

Solving for variable 'l'.

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

Add '3ln' to each side of the equation.
9ln + 3ln + 5lnx = -3ln + 3ln + 8lnx

Combine like terms: 9ln + 3ln = 12ln
12ln + 5lnx = -3ln + 3ln + 8lnx

Combine like terms: -3ln + 3ln = 0
12ln + 5lnx = 0 + 8lnx
12ln + 5lnx = 8lnx

Add '-8lnx' to each side of the equation.
12ln + 5lnx + -8lnx = 8lnx + -8lnx

Combine like terms: 5lnx + -8lnx = -3lnx
12ln + -3lnx = 8lnx + -8lnx

Combine like terms: 8lnx + -8lnx = 0
12ln + -3lnx = 0

Factor out the Greatest Common Factor (GCF), '3ln'.
3ln(4 + -1x) = 0

Ignore the factor 3.
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 + -1x)' equal to zero and attempt to solve:

Simplifying
4 + -1x = 0

Solving
4 + -1x = 0

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

Add '-4' to each side of the equation.
4 + -4 + -1x = 0 + -4

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

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

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

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

Simplifying
0 = -4 + x

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.