Exponential and logarithmic equations
Problem 13.460 · easy
Solve \( \displaystyle e^{x} = 2 \) exactly.
- Divide out the coefficient, then take ln of both sides.
- \[ \left. e^{x} - 2 \right|_{\substack{ x=\ln{\left(2 \right)} }} = 0 \]x = log(2) satisfies the equation.✓ Proved
Answer \( x = \ln{\left(2 \right)} \)
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: fail (misleading) — The first sentence claims to "divide out the coefficient," but the equation e^{x}=2 has no coefficient to divide. This wording could mislead a student into thinking a division step is necessary, though the subsequent logarithm step is correct.qwen3.6:27b-mlx: pass — The solution is correct and concise. The sentence accurately describes the standard method for solving exponential equations, and the equation confirms the result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-01 — The solution is correct and concise. The sentence accurately describes the standard method for solving exponential equations, and the equation confirms the result.gpt-oss:20b: fail (misleading) 2026-11-01 — The first sentence claims to "divide out the coefficient," but the equation e^{x}=2 has no coefficient to divide. This wording could mislead a student into thinking a division step is necessary, though the subsequent logarithm step is correct.qwen3.6:27b-mlx: pass 2026-10-31 — The solution is correct and concise. The sentence accurately describes the method, and the equation verifies the result.gpt-oss:20b: fail (misleading) 2026-10-31 — The first sentence claims to ‘divide out the coefficient’, yet the equation e^{x}=2 has no coefficient to divide. This misleading wording could confuse students about the necessity of a coefficient step. The correct approach is simply to take the natural logarithm of both sides, yielding x=ln(2).qwen3.6:27b-mlx: pass 2026-10-31 — The solution is correct and the steps are logically sound.gpt-oss:20b: fail (error) 2026-10-31 — The first sentence incorrectly states “divide out the coefficient,” but the equation e^{x}=2 has no coefficient to divide. The correct step is simply to take the natural logarithm of both sides.
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 2026-11-01 with SymPy 1.14.0.