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Exponential and logarithmic equations

Problem 13.120 · easy

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

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

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

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 (style) — The sentence 'Divide out the coefficient' is confusing and unnecessary for this specific equation, as there is no coefficient to divide out. It suggests a generic procedure that doesn't apply cleanly here, potentially misleading a student about the necessary steps.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-17 — The sentence 'Divide out the coefficient' is confusing and unnecessary for this specific equation, as there is no coefficient to divide out. It suggests a generic procedure that doesn't apply cleanly here, potentially misleading a student about the necessary steps.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (misleading) 2026-10-14 — The sentence instructs to 'Divide out the coefficient,' but there is no coefficient to divide in e^x = 3. This teaches a false procedural step.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution is correct and concise. The instruction to 'divide out the coefficient' is slightly imprecise phrasing for 'isolating the exponential term', but it does not lead to an error.
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.