Home›Calculus 1›Exponential and logarithmic equations›Problem 13.108

Exponential and logarithmic equations

Problem 13.108 · medium

Solve \( \displaystyle \ln{\left(x \right)} + \ln{\left(x - 2 \right)} = \ln{\left(3 \right)} \) exactly.

Co2bine the logs: ln(x(x − 2)) = ln 3, so x(x − 2) = 3, a quadratic. x = -1 is extraneous: it makes a logarithm's argument negative.
\[ \left. \ln{\left(x \right)} + \ln{\left(x - 2 \right)} - \ln{\left(3 \right)} \right|_{\substack{ x=3 }} = 0 \]
x = 3 satisfies the equation.✓ Proved

Answer \( x = 3 \)

Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	Newton's method over [−20, 20] finds exactly the claimed real roots

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution fails to explicitly state the domain restriction x > 2, which is required for both ln(x) and ln(x-2) to be defined. While it correctly identifies x = -1 as extraneous, it does not explain why x = -1 is the other root of the quadratic (x^2 - 2x - 3 = 0) or rigorously justify that x = 3 is the only valid solution within the domain.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The solution fails to explicitly state the domain restriction x > 2, which is required for both ln(x) and ln(x-2) to be defined. While it correctly identifies x = -1 as extraneous, it does not explain why x = -1 is the other root of the quadratic (x^2 - 2x - 3 = 0) or rigorously justify that x = 3 is the only valid solution within the domain.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly combines logarithms, solves the resulting quadratic, and properly identifies and discards the extraneous root x = -1. The final answer x = 3 is correct and verified.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution claims x = -1 is extraneous without showing that it is actually a root of the derived quadratic equation x^2 - 2x - 3 = 0. The roots are x = 3 and x = -1, but the text skips the step of solving the quadratic to identify -1 as a candidate solution before rejecting it.
gpt-oss:20b: pass 2026-10-14

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/log_exp_equations, checked 2026-10-17 with SymPy 1.14.0.