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Exponential and logarithmic equations

Problem 13.386 · easy

Solve \( \displaystyle 5^{2 x + 1} = 25 \) exactly.

Write 25 as 5^2 and set the exponents equal.
\[ \left. 5^{2 x + 1} - 25 \right|_{\substack{ x=1 \cdot \frac{1}{2} }} = 0 \]
x = 1/2 satisfies the equation.✓ Proved

Answer \( x = \frac{1}{2} \)

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 (error) — The solution skips the crucial algebraic step of solving the linear equation 2x + 1 = 2 for x. It states the method but does not demonstrate the derivation of the final answer.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution skips the crucial algebraic step of solving the linear equation 2x + 1 = 2 for x. It states the method but does not demonstrate the derivation of the final answer.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies that 25 is 5^2, sets the exponents equal (2x + 1 = 2), and solves for x = 1/2. The verification step confirms the result.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies that the bases are equal and sets the exponents equal, leading to the correct exact solution.
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.