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Simplifying ln(2) + -1ln(4x + -1) = ln(2x + 5) Reorder the terms for easier multiplication: 2ln + -1ln(4x + -1) = ln(2x + 5) Reorder the terms: 2ln + -1ln(-1 + 4x) = ln(2x + 5) 2ln + (-1 * -1ln + 4x * -1ln) = ln(2x + 5) 2ln + (1ln + -4lnx) = ln(2x + 5) Combine like terms: 2ln + 1ln = 3ln 3ln + -4lnx = ln(2x + 5) Reorder the terms: 3ln + -4lnx = ln(5 + 2x) 3ln + -4lnx = (5 * ln + 2x * ln) 3ln + -4lnx = (5ln + 2lnx) Solving 3ln + -4lnx = 5ln + 2lnx Solving for variable 'l'. Move all terms containing l to the left, all other terms to the right. Add '-5ln' to each side of the equation. 3ln + -5ln + -4lnx = 5ln + -5ln + 2lnx Combine like terms: 3ln + -5ln = -2ln -2ln + -4lnx = 5ln + -5ln + 2lnx Combine like terms: 5ln + -5ln = 0 -2ln + -4lnx = 0 + 2lnx -2ln + -4lnx = 2lnx Add '-2lnx' to each side of the equation. -2ln + -4lnx + -2lnx = 2lnx + -2lnx Combine like terms: -4lnx + -2lnx = -6lnx -2ln + -6lnx = 2lnx + -2lnx Combine like terms: 2lnx + -2lnx = 0 -2ln + -6lnx = 0 Factor out the Greatest Common Factor (GCF), '-2ln'. -2ln(1 + 3x) = 0 Ignore the factor -2.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 + 3x)' equal to zero and attempt to solve: Simplifying 1 + 3x = 0 Solving 1 + 3x = 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 + 3x = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 3x = 0 + -1 3x = 0 + -1 Combine like terms: 0 + -1 = -1 3x = -1 Add '-3x' to each side of the equation. 3x + -3x = -1 + -3x Combine like terms: 3x + -3x = 0 0 = -1 + -3x Simplifying 0 = -1 + -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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