Home›Calculus 1›Exponential and logarithmic equations›Problem 13.384

Exponential and logarithmic equations

Problem 13.384 · medium

Solve \( \displaystyle \ln{\left(x \right)} + \ln{\left(x - 4 \right)} = \ln{\left(32 \right)} \) exactly.

Co4bine the logs: ln(x(x − 4)) = ln 32, so x(x − 4) = 32, a quadratic. x = -4 is extraneous: it makes a logarithm's argument negative.
Reviewed
\[ \left. \ln{\left(x \right)} + \ln{\left(x - 4 \right)} - \ln{\left(32 \right)} \right|_{\substack{ x=8 }} = 0 \]
x = 8 satisfies the equation.✓ Proved

Answer \( x = 8 \)

Lines: 1 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly combines logarithms, solves the resulting quadratic, and properly identifies and discards the extraneous root x = -4. The final answer x = 8 is correct and verified.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly combines logarithms, solves the resulting quadratic, and properly identifies and discards the extraneous root x = -4. The final answer x = 8 is correct and verified.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution claims x = -4 is extraneous, but x = -4 is not a root of the resulting quadratic equation x^2 - 4x - 32 = 0 (the roots are 8 and -4, wait, (-4)^2 - 4(-4) - 32 = 16 + 16 - 32 = 0, so -4 IS a root). However, the text says 'x = -4 is extraneous' without showing it is a root first, and more importantly, it fails to explicitly solve the quadratic to find the valid root x=8, jumping straight to the check. Actually, looking closer: x(x-4)=32 => x^2-4x-32=0 => (x-8)(x+4)=0. Roots are 8 and -4. -4 is extraneous because ln(-4) is undefined. The sentence says 'x = -4 is extraneous' which is true, but it doesn't explicitly state that x=8 is the other root derived from the quadratic. It just asserts the extraneous nature of -4 and then checks 8. This is logically incomplete as a 'solution' because it doesn't derive x=8, it just verifies it. A student might think they can just guess 8. But is it an 'error'? The prompt asks to solve exactly. The solution combines logs, sets up quadratic, identifies -4 as extraneous, and verifies 8. It misses the step of actually solving the quadratic to get 8. It assumes 8 is the answer. This is a logical gap. It's not just style, it's a missing derivation step.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly solve the quadratic equation x(x-4) = 32 to derive the candidate solutions. It asserts x = -4 is extraneous without showing the calculation that produces it, and it does not demonstrate how x = 8 is obtained, leaving the derivation of the answer incomplete.
gpt-oss:20b: pass 2026-10-18

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 2026-10-19 with SymPy 1.14.0.