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Exponential and logarithmic equations

Problem 13.110 · easy

Solve $ \displaystyle 3^{3 x - 1} = 3 $ exactly.

Write 3 as 3^1 and set the exponents equal.
Reviewed
\[ \left. 3^{3 x - 1} - 3 \right|_{\substack{ x=\frac{2}{3} }} = 0 \]
x = 2/3 satisfies the equation.✓ Proved

Answer $ x = \frac{2}{3} $

✓ 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 applies the property that if $b^u = b^v$, then $u=v$ for $b>0, b eq 1$. The algebraic step of equating exponents is valid and leads to the correct unique solution.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the property that if $b^u = b^v$, then $u=v$ for $b>0, b eq 1$. The algebraic step of equating exponents is valid and leads to the correct unique solution.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the property that if $a^m = a^n$, then $m=n$ for $a>0, a eq 1$. The resulting linear equation is solved correctly.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the one-to-one property of exponential functions with base 3. The algebraic step of equating exponents is valid and leads to the correct exact solution.
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.