Exponential and logarithmic equations
Problem 13.117 · medium
Solve \( \displaystyle 5^{2 x} - 13 \cdot 5^{x} + 36 = 0 \) exactly.
- Let u = 5ˣ: u² − 13u + 36 = 0 factors as (u − 4)(u − 9) = 0.
- \[ \left. 5^{2 x} - 13 \cdot 5^{x} + 36 \right|_{\substack{ x=\frac{2 \ln{\left(2 \right)}}{\ln{\left(5 \right)}} }} = 0 \]x = 2*log(2)/log(5) satisfies the equation.✓ Proved
- \[ \left. 5^{2 x} - 13 \cdot 5^{x} + 36 \right|_{\substack{ x=\frac{2 \ln{\left(3 \right)}}{\ln{\left(5 \right)}} }} = 0 \]x = 2*log(3)/log(5) satisfies the equation.✓ Proved
Answer \( x = \frac{2 \ln{\left(2 \right)}}{\ln{\left(5 \right)}},\ \frac{2 \ln{\left(3 \right)}}{\ln{\left(5 \right)}} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly solve for x from the intermediate results u=4 and u=9. It jumps from the factored form directly to checking the final answers, omitting the critical step of deriving x = log_5(4) and x = log_5(9) and converting them to the required logarithmic form.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to explicitly solve for x from the intermediate results u=4 and u=9. It jumps from the factored form directly to checking the final answers, omitting the critical step of deriving x = log_5(4) and x = log_5(9) and converting them to the required logarithmic form.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution claims the roots are 2*ln(2)/ln(5) and 2*ln(3)/ln(5), which correspond to 5^x = 4 and 5^x = 9. However, the quadratic factors as (u-4)(u-9)=0, so u=4 or u=9. This implies 5^x=4 => x=ln(4)/ln(5) = 2*ln(2)/ln(5) and 5^x=9 => x=ln(9)/ln(5) = 2*ln(3)/ln(5). Wait, the math is actually correct. Let me re-read carefully. u^2 - 13u + 36 = 0. Roots are 4 and 9. 5^x = 4 => x = log_5(4) = ln(4)/ln(5) = 2ln(2)/ln(5). 5^x = 9 => x = log_5(9) = ln(9)/ln(5) = 2ln(3)/ln(5). The stated answer matches the derived answer. The sentence says 'factors as (u-4)(u-9)=0'. This is correct. The equations check out. Why would it fail? Ah, look at the sentence again. 'Let u = 5ˣ: u² − 13u + 36 = 0 factors as (u − 4)(u − 9) = 0.' This sentence asserts the factorization but does not explicitly state the values of u or the subsequent step to find x. However, the subsequent lines verify the specific x values. Is there a missing step? The sentence implies the factorization leads to the answer, but doesn't show the intermediate u values. But more importantly, does the sentence *model* the problem? Yes. Does it lead to the result? Yes. Is it misleading? No. Is it an error? No. Wait, I need to find what is wrong. Let me check the factorization again. (u-4)(u-9) = u^2 - 9u - 4u + 36 = u^2 - 13u + 36. Correct. The roots are 4 and 9. 5^x = 4 => x = log_5(4). 5^x = 9 => x = log_5(9). The answer provided is 2ln(2)/ln(5) and 2ln(3)/ln(5). These are correct. Why would this be a fail? Perhaps the sentence is considered incomplete because it doesn't explicitly solve for u? Or perhaps the prompt implies I should find an error where there is none? No, the instructions say 'find what is wrong'. If there is nothing wrong, I should say pass. But usually these tasks have a subtle error. Let me look really closely at the syntax. 'Subs(...)' is a CAS command. The sentence is 'Let u = 5ˣ: u² − 13u + 36 = 0 factors as (u − 4)(u − 9) = 0.' This is a valid mathematical statement. The equations prove the answers. Is there a domain issue? 5^x is always positive, 4 and 9 are positive, so real solutions exist. Everything seems correct. Let me re-read the 'Stated answer'. x = 2 ln(2) / ln(5), 2 ln(3) / ln(5). This is correct. Is it possible the user made a typo in the problem statement? 5^{2x} - 13*5^x + 36 = 0. If the constant was different, say 32, roots would be different. But 36 factors to 4 and 9. 4+9=13. It's consistent. I will mark it as pass because I cannot find an error.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the substitution, solves the quadratic, and provides the exact logarithmic forms for the roots. The verification steps confirm the results.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.