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Exponential and logarithmic equations

Problem 13.459 · medium

Solve \( \displaystyle \ln{\left(x \right)} + \ln{\left(x - 4 \right)} = \ln{\left(21 \right)} \) exactly.

Co4bine the logs: ln(x(x − 4)) = ln 21, so x(x − 4) = 21, a quadratic. x = -3 is extraneous: it makes a logarithm's argument negative.
\[ \left. \ln{\left(x \right)} + \ln{\left(x - 4 \right)} - \ln{\left(21 \right)} \right|_{\substack{ x=7 }} = 0 \]
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 (error) — The solution claims x = -3 is a root of the quadratic x(x-4)=21, but the roots are actually 7 and -3. While -3 is indeed extraneous, the solution fails to explicitly derive or state the correct roots, jumping from the quadratic form to the rejection of -3 without showing that 7 is the other root. More critically, it does not explicitly solve the quadratic equation x^2 - 4x - 21 = 0 to find x=7 and x=-3, making the logic incomplete and potentially confusing for a student who might not know where 7 came from other than the final check.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-01 — The solution claims x = -3 is a root of the quadratic x(x-4)=21, but the roots are actually 7 and -3. While -3 is indeed extraneous, the solution fails to explicitly derive or state the correct roots, jumping from the quadratic form to the rejection of -3 without showing that 7 is the other root. More critically, it does not explicitly solve the quadratic equation x^2 - 4x - 21 = 0 to find x=7 and x=-3, making the logic incomplete and potentially confusing for a student who might not know where 7 came from other than the final check.
gpt-oss:20b: pass 2026-11-01
qwen3.6:27b-mlx: pass 2026-10-31 — The solution correctly applies logarithm properties, identifies the valid root, and rejects the extraneous solution. The verification step confirms the result.
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 or verify the domain constraints (x > 4) required for the logarithms to be defined, jumping straight to discarding the negative root. This omits the crucial step of ensuring the positive root (x=7) actually lies within the valid domain, which is necessary for a complete and rigorous solution.
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.