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

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


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

Multiply ln * x
lnx + ln(x + -3) = ln(6x) + -1ln(x + -2)

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

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

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

Remove parenthesis around (6x)
-3ln + 2lnx = ln * 6x + -1ln(x + -2)

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

Multiply ln * x
-3ln + 2lnx = 6lnx + -1ln(x + -2)

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

Reorder the terms:
-3ln + 2lnx = 2ln + 6lnx + -1lnx

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

Solving
-3ln + 2lnx = 2ln + 5lnx

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.
-3ln + -2ln + 2lnx = 2ln + -2ln + 5lnx

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

Combine like terms: 2ln + -2ln = 0
-5ln + 2lnx = 0 + 5lnx
-5ln + 2lnx = 5lnx

Add '-5lnx' to each side of the equation.
-5ln + 2lnx + -5lnx = 5lnx + -5lnx

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

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

Factor out the Greatest Common Factor (GCF), '-1ln'.
-1ln(5 + 3x) = 0

Ignore the factor -1.

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 '(5 + 3x)' equal to zero and attempt to solve: Simplifying 5 + 3x = 0 Solving 5 + 3x = 0 Move all terms containing l to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + 3x = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 3x = 0 + -5 3x = 0 + -5 Combine like terms: 0 + -5 = -5 3x = -5 Add '-3x' to each side of the equation. 3x + -3x = -5 + -3x Combine like terms: 3x + -3x = 0 0 = -5 + -3x Simplifying 0 = -5 + -3x 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.

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