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Exponential and logarithmic equations

Problem 13.189 · easy

Solve \( \displaystyle 2 e^{2 x} = 10 \) exactly.

Divide out the coefficient, then take ln of both sides.
\[ \left. 2 e^{2 x} - 10 \right|_{\substack{ x=\frac{\ln{\left(5 \right)}}{2} }} = 0 \]
x = log(5)/2 satisfies the equation.✓ Proved

Answer \( x = \frac{\ln{\left(5 \right)}}{2} \)

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 (misleading) — The solution only verifies that the candidate is a solution, but does not demonstrate that it is the unique solution. It fails to explain why taking the natural logarithm is valid (requiring the argument to be positive) or why the solution is unique.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (misleading) 2026-10-18 — The solution only verifies that the candidate is a solution, but does not demonstrate that it is the unique solution. It fails to explain why taking the natural logarithm is valid (requiring the argument to be positive) or why the solution is unique.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the algebraic steps (isolating the exponential term and applying the natural logarithm) and verifies the result. The logic is sound and complete for the scope of the problem.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the algebraic steps required to isolate x and provides the exact solution. The verification step confirms the result.
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 2026-10-18 with SymPy 1.14.0.