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Exponential and logarithmic equations

Problem 13.282 · easy

Solve \( \displaystyle 5^{3 x - 2} = 125 \) exactly.

Write 125 as 5^3 and set the exponents equal.
Reviewed
\[ \left. 5^{3 x - 2} - 125 \right|_{\substack{ x=\frac{5}{3} }} = 0 \]
x = 5/3 satisfies the equation.✓ Proved

Answer \( x = \frac{5}{3} \)

✓ 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 identifies that 125 is 5^3 and equates the exponents to solve for x. The verification step confirms the result.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies that 125 is 5^3 and equates the exponents to solve for x. The verification step confirms the result.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies that 125 is 5^3, sets the exponents equal (3x - 2 = 3), and solves for x = 5/3. The verification step confirms the result.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies that 125 is 5^3 and equates the exponents, leading to the correct exact solution. The verification step confirms the result.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.