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Exponential and logarithmic equations

Problem 13.468 · medium

Solve \( \displaystyle \ln{\left(x \right)} + \ln{\left(x - 2 \right)} = \ln{\left(24 \right)} \) exactly.

Co2bine the logs: ln(x(x − 2)) = ln 24, so x(x − 2) = 24, a quadratic. x = -4 is extraneous: it makes a logarithm's argument negative.
\[ \left. \ln{\left(x \right)} + \ln{\left(x - 2 \right)} - \ln{\left(24 \right)} \right|_{\substack{ x=6 }} = 0 \]
x = 6 satisfies the equation.✓ Proved

Answer \( x = 6 \)

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 = -4 is a root of the quadratic x(x-2)=24, but the roots are actually 6 and -4. While -4 is indeed extraneous, the solution fails to explicitly derive or state the correct roots (6 and -4) before discarding the extraneous one, making the logical jump from the quadratic to the final answer opaque and potentially confusing regarding which root was kept.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-01 — The solution claims x = -4 is a root of the quadratic x(x-2)=24, but the roots are actually 6 and -4. While -4 is indeed extraneous, the solution fails to explicitly derive or state the correct roots (6 and -4) before discarding the extraneous one, making the logical jump from the quadratic to the final answer opaque and potentially confusing regarding which root was kept.
gpt-oss:20b: pass 2026-11-01
qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution claims x = -4 is a root of the quadratic x(x-2)=24, but the roots are actually x=6 and x=-4. While x=-4 is indeed extraneous, the solution fails to explicitly derive or state the correct roots, jumping from the quadratic form to the rejection of -4 without showing how 6 was obtained or that -4 was a candidate. More critically, it implies x=-4 is a solution to the quadratic without showing the factorization (x-6)(x+4)=0, making the logic opaque and potentially confusing for a student who might not see where -4 came from or why it's the only other option.
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: fail (style) 2026-10-31 — [domain objection, downgraded to style] The solution claims x = -4 is extraneous, but x = -4 is not a root of the quadratic equation x^2 - 2x - 24 = 0 (the roots are 6 and -4? No, 6*-4 = -24, 6-4=2 != -2. The roots are 6 and -4? Let's check: 6*4=24, 6-4=2. Wait. x(x-2)=24 => x^2-2x-24=0. (x-6)(x+4)=0. Roots are 6 and -4. Okay, so -4 IS a root of the quadratic. The text says 'x = -4 is extraneous'. This is correct. However, the text says 'x = -4 is extraneous: it makes a logarithm's argument negative.' This is correct reasoning. Why did I think it was wrong? Let me re-read carefully. 'x = -4 is extraneous: it makes a logarithm's argument negative.' This is true. ln(-4) is undefined. So the sentence is actually correct. Is there another error? 'Co2bine' is a typo. 'Subs(...)' is an equation line. The sentence says 'x = -4 is extraneous'. It does not explicitly state the other root is 6, but implies it by discarding -4. Is it misleading? It skips the step of finding the roots. It jumps from 'quadratic' to 'x=-4 is extraneous'. A student might not know where -4 came from. But the main issue is usually clarity. Let's look closer. 'x(x-2)=24, a quadratic. x = -4 is extraneous'. It doesn't say 'The roots are 6 and -4'. It just asserts -4 is extraneous. This is slightly confusing but not strictly wrong. However, looking at the provided answer 'x=6', and the check 'x=6 satisfies', the logic holds. Is there a hidden error? The prompt asks to find what is WRONG. 'Co2bine' is a style error. Is there a mathematical error? No. Is it misleading? It omits the calculation of the roots. A student might think -4 is the only other option without knowing how to get it. But typically 'error' is reserved for math mistakes. 'Misleading' for bad pedagogy. Skipping the quadratic formula/factoring step is bad pedagogy. But is it 'misleading'? It might lead a student to think they don't need to solve the quadratic. Let's look for a harder error. Did I miss a root? x^2 - 2x - 24 = 0. Discriminant 4 + 96 = 100. Roots (2 +/- 10)/2. 12/2 = 6. -8/2 = -4. Correct. So -4 is indeed the other root. The sentence is mathematically sound but pedagogically poor because it skips the solution of the quadratic. However, usually 'style' covers typos and minor omissions. 'Misleading' covers things that teach false concepts. Does it teach a false concept? No. It just skips a step. Wait, look at the first sentence again. 'Co2bine the logs... so x(x-2)=24, a quadratic. x = -4 is extraneous'. It presents -4 out of thin air. This is a 'style' or 'misleading' issue depending on strictness. Let's look at the severity definitions. 'Misleading: a student would learn something false'. Would a student learn something false? They might learn that you can just guess the extraneous root? No. They might just be confused. 'Style: correct, but a sentence could be clearer'. This fits best. But wait, is there an error in the equation line? 'Subs(..., x, 6) = 0'. This is correct. Let's re-read the prompt constraints. 'Find what is wrong'. If it's just a typo and a skipped step, is it a fail? Yes, 'pass' is for 'sound'. Is skipping the derivation of the roots 'sound'? It's incomplete. Incomplete solutions are often 'fail' in these adversarial reviews if they don't justify the result. The result x=6 is not derived, it's just checked. The derivation of x=6 is missing. The text says 'x=-4 is extraneous', implying the other root is the answer, but never states the other root is 6. It relies on the reader to know the roots are 6 and -4. This is a gap in the logic chain. The conclusion 'x=6' is stated in the problem's 'Stated answer', but the solution text must derive it. The solution text does NOT derive x=6. It only checks it. Therefore, the solution does not actually solve the problem, it only verifies the answer. This is a major logical gap. The sentence 'x = -4 is extraneous' does not lead to 'x = 6' without the intermediate step 'The other root is 6'. Thus, the conclusion does not follow from the lines before it. Verdict: Fail. Severity: Error (logical gap/missing step).
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.