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Exponential and logarithmic equations

Problem 13.286 · medium

Solve \( \displaystyle 2^{2 x} - 10 \cdot 2^{x} + 9 = 0 \) exactly.

Let u = 2ˣ: u² − 10u + 9 = 0 factors as (u − 1)(u − 9) = 0.
\[ \left. 2^{2 x} - 10 \cdot 2^{x} + 9 \right|_{\substack{ x=0 }} = 0 \]
x = 0 satisfies the equation.✓ Proved
\[ \left. 2^{2 x} - 10 \cdot 2^{x} + 9 \right|_{\substack{ x=\frac{2 \ln{\left(3 \right)}}{\ln{\left(2 \right)}} }} = 0 \]
x = 2*log(3)/log(2) satisfies the equation.✓ Proved

Answer \( x = 0,\ \frac{2 \ln{\left(3 \right)}}{\ln{\left(2 \right)}} \)

Lines: 2 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
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: fail (error) — The solution fails to solve for x from the second factor u=9. It states u=9 but then jumps to verifying x = 2*ln(3)/ln(2) without showing the derivation x = log_2(9) = 2*log_2(3). While the verified answer is correct, the logical step connecting u=9 to the final x value is missing, making the solution incomplete and potentially confusing regarding how the second root was obtained.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to solve for x from the second factor u=9. It states u=9 but then jumps to verifying x = 2*ln(3)/ln(2) without showing the derivation x = log_2(9) = 2*log_2(3). While the verified answer is correct, the logical step connecting u=9 to the final x value is missing, making the solution incomplete and potentially confusing regarding how the second root was obtained.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to solve for x after finding u = 1 and u = 9. It lists the correct answers but does not show the derivation (x = 0 and x = log2(9)), making the logic incomplete.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to solve for x from the second factor u=9. It correctly identifies u=1 and u=9, but only checks x=0 (from u=1) and skips deriving x=log_2(9) from u=9, instead just verifying the final answer without showing the derivation.
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.