Home›Calculus 1›Exponential and logarithmic equations›Problem 13.287

Exponential and logarithmic equations

Problem 13.287 · medium

Solve \( \displaystyle \ln{\left(x \right)} + \ln{\left(x - 1 \right)} = \ln{\left(2 \right)} \) exactly.

Co1bine the logs: ln(x(x − 1)) = ln 2, so x(x − 1) = 2, a quadratic. x = -1 is extraneous: it makes a logarithm's argument negative.
\[ \left. \ln{\left(x \right)} + \ln{\left(x - 1 \right)} - \ln{\left(2 \right)} \right|_{\substack{ x=2 }} = 0 \]
x = 2 satisfies the equation.✓ Proved

Answer \( x = 2 \)

Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 claims x = -1 is a solution to the quadratic but is extraneous. However, x = -1 is not a root of x(x-1)=2 (which yields x^2-x-2=0, roots 2 and -1). Wait, x=-1 IS a root of x^2-x-2=0. (-1)(-2) = 2. So x=-1 is a root of the quadratic. The sentence says 'x = -1 is extraneous'. This is correct reasoning. But the sentence combines the setup, solving, and checking into one messy block. More importantly, it fails to explicitly state the domain constraint x > 1 derived from the original equation (ln(x) and ln(x-1) require x>0 and x>1). While it identifies -1 as extraneous, it does not explain *why* based on the domain, just that it makes an argument negative. This is acceptable. However, looking closer: 'x = -1 is extraneous: it makes a logarithm's argument negative.' This is true. Is there a better issue? The prompt asks to solve exactly. The solution finds x=2. It checks x=2. It discards x=-1. The logic is sound. Why would it fail? Let's re-read carefully. 'x(x-1)=2, a quadratic. x = -1 is extraneous...'. It skips the step of actually solving the quadratic to find the roots 2 and -1. It jumps from 'it's a quadratic' to 'x=-1 is extraneous'. A student might not know how to get -1. It implies -1 is the other root without showing the factorization or quadratic formula. This is a 'style' or 'misleading' issue because it hides the derivation of the candidate solutions. It asserts x=-1 is a root without showing work. Actually, the biggest issue is that it doesn't explicitly solve for x. It says 'so x(x-1)=2, a quadratic.' Then immediately 'x=-1 is extraneous'. It never states 'The roots are 2 and -1'. It assumes the reader knows the roots. This is poor pedagogy. Also, strictly speaking, ln(x) + ln(x-1) requires x > 1. The solution only checks that -1 makes an argument negative. It doesn't establish the domain x > 1 which would immediately rule out -1 and confirm 2 is the only candidate. Let's look for a harder error. Did it miss a solution? No. Did it include a wrong one? No. Is the sentence 'x = -1 is extraneous: it makes a logarithm's argument negative' technically correct? Yes. Is the omission of the solving step 'error'? No, but it's 'misleading' or 'style'. Let's look at the JSON format. Wait, look at the first sentence again. 'Co1bine the logs... so x(x-1)=2... x=-1 is extraneous'. It never explicitly states that x=2 is the *other* root. It just checks x=2 in the next line. It's a bit disjointed. However, often these adversarial reviews look for mathematical errors. Is x=-1 really a root? (-1)^2 - (-1) - 2 = 1 + 1 - 2 = 0. Yes. Is there any other root? No. So the math is correct. The presentation is sloppy. Let's consider if 'misleading' is the right severity. A student might think 'Oh, I just need to guess roots and check them'. Or they might not know how to solve the quadratic. Let's try to find a stricter error. The problem asks to solve *exactly*. The solution says 'x = -1 is extraneous'. It does not say 'The solutions to the quadratic are 2 and -1'. I will mark it as 'misleading' because it skips the crucial step of finding the roots of the quadratic, potentially teaching students that they don't need to solve the quadratic, just check random numbers or that -1 is somehow obvious. Actually, looking at similar examples, skipping the solution of the quadratic is usually considered a 'style' or 'minor' issue unless it's a 'show your work' context. But here, the instruction is 'find what is wrong'. Let's look at the domain. ln(x) and ln(x-1). Domain is x > 1. The solution says 'x=-1 is extraneous'. It doesn't mention x=2 is valid because it's in the domain. It just checks it satisfies the equation. I'll go with 'misleading' because it obscures the method for finding the roots.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution claims x = -1 is a solution to the quadratic but is extraneous. However, x = -1 is not a root of x(x-1)=2 (which yields x^2-x-2=0, roots 2 and -1). Wait, x=-1 IS a root of x^2-x-2=0. (-1)(-2) = 2. So x=-1 is a root of the quadratic. The sentence says 'x = -1 is extraneous'. This is correct reasoning. But the sentence combines the setup, solving, and checking into one messy block. More importantly, it fails to explicitly state the domain constraint x > 1 derived from the original equation (ln(x) and ln(x-1) require x>0 and x>1). While it identifies -1 as extraneous, it does not explain *why* based on the domain, just that it makes an argument negative. This is acceptable. However, looking closer: 'x = -1 is extraneous: it makes a logarithm's argument negative.' This is true. Is there a better issue? The prompt asks to solve exactly. The solution finds x=2. It checks x=2. It discards x=-1. The logic is sound. Why would it fail? Let's re-read carefully. 'x(x-1)=2, a quadratic. x = -1 is extraneous...'. It skips the step of actually solving the quadratic to find the roots 2 and -1. It jumps from 'it's a quadratic' to 'x=-1 is extraneous'. A student might not know how to get -1. It implies -1 is the other root without showing the factorization or quadratic formula. This is a 'style' or 'misleading' issue because it hides the derivation of the candidate solutions. It asserts x=-1 is a root without showing work. Actually, the biggest issue is that it doesn't explicitly solve for x. It says 'so x(x-1)=2, a quadratic.' Then immediately 'x=-1 is extraneous'. It never states 'The roots are 2 and -1'. It assumes the reader knows the roots. This is poor pedagogy. Also, strictly speaking, ln(x) + ln(x-1) requires x > 1. The solution only checks that -1 makes an argument negative. It doesn't establish the domain x > 1 which would immediately rule out -1 and confirm 2 is the only candidate. Let's look for a harder error. Did it miss a solution? No. Did it include a wrong one? No. Is the sentence 'x = -1 is extraneous: it makes a logarithm's argument negative' technically correct? Yes. Is the omission of the solving step 'error'? No, but it's 'misleading' or 'style'. Let's look at the JSON format. Wait, look at the first sentence again. 'Co1bine the logs... so x(x-1)=2... x=-1 is extraneous'. It never explicitly states that x=2 is the *other* root. It just checks x=2 in the next line. It's a bit disjointed. However, often these adversarial reviews look for mathematical errors. Is x=-1 really a root? (-1)^2 - (-1) - 2 = 1 + 1 - 2 = 0. Yes. Is there any other root? No. So the math is correct. The presentation is sloppy. Let's consider if 'misleading' is the right severity. A student might think 'Oh, I just need to guess roots and check them'. Or they might not know how to solve the quadratic. Let's try to find a stricter error. The problem asks to solve *exactly*. The solution says 'x = -1 is extraneous'. It does not say 'The solutions to the quadratic are 2 and -1'. I will mark it as 'misleading' because it skips the crucial step of finding the roots of the quadratic, potentially teaching students that they don't need to solve the quadratic, just check random numbers or that -1 is somehow obvious. Actually, looking at similar examples, skipping the solution of the quadratic is usually considered a 'style' or 'minor' issue unless it's a 'show your work' context. But here, the instruction is 'find what is wrong'. Let's look at the domain. ln(x) and ln(x-1). Domain is x > 1. The solution says 'x=-1 is extraneous'. It doesn't mention x=2 is valid because it's in the domain. It just checks it satisfies the equation. I'll go with 'misleading' because it obscures the method for finding the roots.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The solution fails to explicitly state the domain constraints (x > 1) required for the logarithms to be defined. While it correctly identifies x = -1 as extraneous, it does not explain why x = 2 is the *only* valid solution, nor does it verify that x = 2 satisfies the domain constraints. A rigorous solution must establish the domain before or during the solving process.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The solution fails to explicitly state the domain restriction x > 1 required for the original logarithmic expressions to be defined. While it correctly identifies x = -1 as extraneous, it does not explain why the other root of the quadratic (x = -1 is one, the other is 2) is the only valid one, nor does it verify that x=2 satisfies the domain constraint x > 1. A rigorous solution must establish the domain before or during the solving process to ensure validity.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.