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Exponential and logarithmic equations

Problem 13.111 · easy

Solve \( \displaystyle \ln_{3}\left(4 x + 2\right) = 1 \) exactly.

Rewrite in exponential form: 4*x + 2 = 3^1.
Reviewed
\[ \left. \frac{\ln{\left(4 x + 2 \right)}}{\ln{\left(3 \right)}} - 1 \right|_{\substack{ x=1 \cdot \frac{1}{4} }} = 0 \]
x = 1/4 satisfies the equation.✓ Proved

Answer \( x = \frac{1}{4} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. 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	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 converts the logarithmic equation to its exponential form and verifies the solution. The steps are logically sound and complete.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly converts the logarithmic equation to its exponential form and verifies the solution. The steps are logically sound and complete.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly converts the logarithmic equation to its exponential form and solves for x. The verification step confirms the solution is valid.
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.