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Exponential and logarithmic equations

Problem 13.112 · medium

Solve \( \displaystyle 5^{2 x} - 4 \cdot 5^{x} + 3 = 0 \) exactly.

Let u = 5ˣ: u² − 4u + 3 = 0 factors as (u − 3)(u − 1) = 0.
Reviewed
\[ \left. 5^{2 x} - 4 \cdot 5^{x} + 3 \right|_{\substack{ x=0 }} = 0 \]
x = 0 satisfies the equation.✓ Proved
\[ \left. 5^{2 x} - 4 \cdot 5^{x} + 3 \right|_{\substack{ x=\frac{\ln{\left(3 \right)}}{\ln{\left(5 \right)}} }} = 0 \]
x = log(3)/log(5) satisfies the equation.✓ Proved

Answer \( x = 0,\ \frac{\ln{\left(3 \right)}}{\ln{\left(5 \right)}} \)

✓ Nihil obstat Lines: 2 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
3	✓ 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 the substitution, solves the resulting quadratic, and verifies the exact solutions.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the substitution, solves the resulting quadratic, and verifies the exact solutions.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the substitution, solves the quadratic, and verifies the roots. Although it omits the explicit derivation of x from u, the verification steps confirm the stated answer is correct and complete.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the substitution, solves the quadratic, and verifies the resulting values for x.
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.