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Exponential and logarithmic equations

Problem 13.119 · easy

Solve \( \displaystyle \ln_{5}\left(2 x - 3\right) = 0 \) exactly.

Rewrite in exponential form: 2*x - 3 = 5^0.
\[ \left. \frac{\ln{\left(2 x - 3 \right)}}{\ln{\left(5 \right)}} \right|_{\substack{ x=2 }} = 0 \]
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. 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 is incomplete because it only verifies the candidate answer rather than deriving it. A proper solution must show the algebraic steps to isolate x (i.e., 2x - 3 = 1 => 2x = 4 => x = 2) and ideally check the domain constraint (2x - 3 > 0).

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The solution is incomplete because it only verifies the candidate answer rather than deriving it. A proper solution must show the algebraic steps to isolate x (i.e., 2x - 3 = 1 => 2x = 4 => x = 2) and ideally check the domain constraint (2x - 3 > 0).
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (misleading) 2026-10-14 — The solution verifies that x=2 is a solution but fails to demonstrate that it is the unique solution. Since the logarithmic function is one-to-one, the step from the exponential form to the final answer should explicitly state uniqueness or solve for x algebraically, rather than just checking a candidate.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the definition of logarithms to convert the equation to exponential form and solves for x. The verification step confirms the result.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.