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Exponential and logarithmic equations

Problem 13.467 · easy

Solve \( \displaystyle 10^{3 x + 1} = 100 \) exactly.

Write 100 as 10^2 and set the exponents equal.
Reviewed
\[ \left. 10^{3 x + 1} - 100 \right|_{\substack{ x=1 \cdot \frac{1}{3} }} = 0 \]
x = 1/3 satisfies the equation.✓ Proved

Answer \( x = \frac{1}{3} \)

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 identifies that equating exponents is valid because the bases are identical and non-zero. The final answer is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-01 — The solution correctly identifies that equating exponents is valid because the bases are identical and non-zero. The final answer is correct.
gpt-oss:20b: pass 2026-11-01
qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution skips the crucial algebraic step of solving the resulting linear equation 3x + 1 = 2 for x. It jumps from setting exponents equal to the final answer without showing the derivation, which is a logical gap in a 'solve' problem.
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: pass 2026-10-31 — The solution correctly applies the one-to-one property of exponential functions with base 10. The setup is sound and the conclusion follows logically.
gpt-oss:20b: pass 2026-10-31

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 2026-11-01 with SymPy 1.14.0.