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Simplifying ln(5x + -2) + -1ln(3x + -2) = ln(15x + -6) + -1ln(5x + 2) Reorder the terms: ln(-2 + 5x) + -1ln(3x + -2) = ln(15x + -6) + -1ln(5x + 2) (-2 * ln + 5x * ln) + -1ln(3x + -2) = ln(15x + -6) + -1ln(5x + 2) (-2ln + 5lnx) + -1ln(3x + -2) = ln(15x + -6) + -1ln(5x + 2) Reorder the terms: -2ln + 5lnx + -1ln(-2 + 3x) = ln(15x + -6) + -1ln(5x + 2) -2ln + 5lnx + (-2 * -1ln + 3x * -1ln) = ln(15x + -6) + -1ln(5x + 2) -2ln + 5lnx + (2ln + -3lnx) = ln(15x + -6) + -1ln(5x + 2) Reorder the terms: -2ln + 2ln + 5lnx + -3lnx = ln(15x + -6) + -1ln(5x + 2) Combine like terms: -2ln + 2ln = 0 0 + 5lnx + -3lnx = ln(15x + -6) + -1ln(5x + 2) 5lnx + -3lnx = ln(15x + -6) + -1ln(5x + 2) Combine like terms: 5lnx + -3lnx = 2lnx 2lnx = ln(15x + -6) + -1ln(5x + 2) Reorder the terms: 2lnx = ln(-6 + 15x) + -1ln(5x + 2) 2lnx = (-6 * ln + 15x * ln) + -1ln(5x + 2) 2lnx = (-6ln + 15lnx) + -1ln(5x + 2) Reorder the terms: 2lnx = -6ln + 15lnx + -1ln(2 + 5x) 2lnx = -6ln + 15lnx + (2 * -1ln + 5x * -1ln) 2lnx = -6ln + 15lnx + (-2ln + -5lnx) Reorder the terms: 2lnx = -6ln + -2ln + 15lnx + -5lnx Combine like terms: -6ln + -2ln = -8ln 2lnx = -8ln + 15lnx + -5lnx Combine like terms: 15lnx + -5lnx = 10lnx 2lnx = -8ln + 10lnx Solving 2lnx = -8ln + 10lnx Solving for variable 'l'. Move all terms containing l to the left, all other terms to the right. Add '8ln' to each side of the equation. 8ln + 2lnx = -8ln + 8ln + 10lnx Combine like terms: -8ln + 8ln = 0 8ln + 2lnx = 0 + 10lnx 8ln + 2lnx = 10lnx Add '-10lnx' to each side of the equation. 8ln + 2lnx + -10lnx = 10lnx + -10lnx Combine like terms: 2lnx + -10lnx = -8lnx 8ln + -8lnx = 10lnx + -10lnx Combine like terms: 10lnx + -10lnx = 0 8ln + -8lnx = 0 Factor out the Greatest Common Factor (GCF), '8ln'. 8ln(1 + -1x) = 0 Ignore the factor 8.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 '(1 + -1x)' equal to zero and attempt to solve: Simplifying 1 + -1x = 0 Solving 1 + -1x = 0 Move all terms containing l to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + -1x = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -1x = 0 + -1 -1x = 0 + -1 Combine like terms: 0 + -1 = -1 -1x = -1 Add 'x' to each side of the equation. -1x + x = -1 + x Combine like terms: -1x + x = 0 0 = -1 + x Simplifying 0 = -1 + 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.
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