ln(x-3)+ln(12x+24)=ln(-2x)

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Solution for ln(x-3)+ln(12x+24)=ln(-2x) equation:


Simplifying
ln(x + -3) + ln(12x + 24) = ln(-2x)

Reorder the terms:
ln(-3 + x) + ln(12x + 24) = ln(-2x)
(-3 * ln + x * ln) + ln(12x + 24) = ln(-2x)
(-3ln + lnx) + ln(12x + 24) = ln(-2x)

Reorder the terms:
-3ln + lnx + ln(24 + 12x) = ln(-2x)
-3ln + lnx + (24 * ln + 12x * ln) = ln(-2x)
-3ln + lnx + (24ln + 12lnx) = ln(-2x)

Reorder the terms:
-3ln + 24ln + lnx + 12lnx = ln(-2x)

Combine like terms: -3ln + 24ln = 21ln
21ln + lnx + 12lnx = ln(-2x)

Combine like terms: lnx + 12lnx = 13lnx
21ln + 13lnx = ln(-2x)

Remove parenthesis around (-2x)
21ln + 13lnx = ln * -2x

Reorder the terms for easier multiplication:
21ln + 13lnx = -2ln * x

Multiply ln * x
21ln + 13lnx = -2lnx

Solving
21ln + 13lnx = -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.
21ln + 13lnx + 2lnx = -2lnx + 2lnx

Combine like terms: 13lnx + 2lnx = 15lnx
21ln + 15lnx = -2lnx + 2lnx

Combine like terms: -2lnx + 2lnx = 0
21ln + 15lnx = 0

Factor out the Greatest Common Factor (GCF), '3ln'.
3ln(7 + 5x) = 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 '(7 + 5x)' equal to zero and attempt to solve: Simplifying 7 + 5x = 0 Solving 7 + 5x = 0 Move all terms containing l to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + 5x = 0 + -7 Combine like terms: 7 + -7 = 0 0 + 5x = 0 + -7 5x = 0 + -7 Combine like terms: 0 + -7 = -7 5x = -7 Add '-5x' to each side of the equation. 5x + -5x = -7 + -5x Combine like terms: 5x + -5x = 0 0 = -7 + -5x Simplifying 0 = -7 + -5x 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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