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Exponential and logarithmic equations

Problem 13.294 · easy

Solve \( \displaystyle 2 e^{3 x} = 4 \) exactly.

Divide out the coefficient, then take ln of both sides.
Reviewed
\[ \left. 2 e^{3 x} - 4 \right|_{\substack{ x=\frac{\ln{\left(2 \right)}}{3} }} = 0 \]
x = log(2)/3 satisfies the equation.✓ Proved

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

✓ 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 the algebraic steps (isolating the exponential term and applying the natural logarithm) and verifies the result. The logic is sound and complete for an exact solution.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — 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 an exact solution.
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.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution is correct and the steps are logically sound.
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.