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Exponential and logarithmic equations

Problem 13.461 · medium

Solve \( \displaystyle \ln{\left(x \right)} + \ln{\left(x - 3 \right)} = \ln{\left(28 \right)} \) exactly.

Co3bine the logs: ln(x(x − 3)) = ln 28, so x(x − 3) = 28, a quadratic. x = -4 is extraneous: it makes a logarithm's argument negative.
\[ \left. \ln{\left(x \right)} + \ln{\left(x - 3 \right)} - \ln{\left(28 \right)} \right|_{\substack{ x=7 }} = 0 \]
x = 7 satisfies the equation.✓ Proved

Answer \( x = 7 \)

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 > 3, which is required for both logarithms to be defined. While it correctly identifies x = -4 as extraneous, it does not explain why x = 7 is the only valid solution (i.e., that it satisfies x > 3), leaving a gap in the logical justification.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-11-01 — [domain objection, downgraded to style] The solution fails to explicitly state the domain restriction x > 3, which is required for both logarithms to be defined. While it correctly identifies x = -4 as extraneous, it does not explain why x = 7 is the only valid solution (i.e., that it satisfies x > 3), leaving a gap in the logical justification.
gpt-oss:20b: pass 2026-11-01
qwen3.6:27b-mlx: fail (misleading) 2026-10-31 — The solution claims x = -4 is extraneous without showing it is a root of the derived quadratic equation x^2 - 3x - 28 = 0. A student might incorrectly assume -4 is a solution to the quadratic or miss the step of solving for the roots entirely.
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: fail (style) 2026-10-31 — [domain objection, downgraded to style] The solution fails to explicitly state the domain constraints (x > 3) required for the logarithms to be defined. While it correctly identifies x = -4 as extraneous, it does not explain why x = 7 is the only valid solution among the roots of the quadratic, nor does it verify that x = 7 satisfies the domain constraint.
gpt-oss:20b: pass 2026-10-31

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 2026-11-01 with SymPy 1.14.0.