ln(x-5)=ln(x-3)-ln(x-1)

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


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

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

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

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

Reorder the terms:
-5ln + lnx = -3ln + 1ln + lnx + -1lnx

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

Combine like terms: lnx + -1lnx = 0
-5ln + lnx = -2ln + 0
-5ln + lnx = -2ln

Solving
-5ln + lnx = -2ln

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

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

Combine like terms: -2ln + 2ln = 0
-3ln + lnx = 0

Factor out the Greatest Common Factor (GCF), 'ln'.
ln(-3 + 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 '(-3 + x)' equal to zero and attempt to solve: Simplifying -3 + x = 0 Solving -3 + x = 0 Move all terms containing l to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = 0 + 3 Combine like terms: -3 + 3 = 0 0 + x = 0 + 3 x = 0 + 3 Combine like terms: 0 + 3 = 3 x = 3 Add '-1x' to each side of the equation. x + -1x = 3 + -1x Combine like terms: x + -1x = 0 0 = 3 + -1x Simplifying 0 = 3 + -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.

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