Exponential and logarithmic equations
Problem 13.184 · easy
Solve \( \displaystyle e^{2 x} = 3 \) exactly.
- Divide out the coefficient, then take ln of both sides.
- \[ \left. e^{2 x} - 3 \right|_{\substack{ x=\frac{\ln{\left(3 \right)}}{2} }} = 0 \]x = log(3)/2 satisfies the equation.✓ Proved
Answer \( x = \frac{\ln{\left(3 \right)}}{2} \)
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: fail (misleading) — The first sentence incorrectly states that a coefficient is divided out; the equation e^{2x}=3 has no coefficient to remove. The correct step is simply taking the natural logarithm of both sides.qwen3.6:27b-mlx: fail (error) — The solution only verifies that the candidate is a root, but does not prove it is the unique solution. Since the exponential function is strictly monotonic, uniqueness must be established to fully 'solve' the equation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution only verifies that the candidate is a root, but does not prove it is the unique solution. Since the exponential function is strictly monotonic, uniqueness must be established to fully 'solve' the equation.gpt-oss:20b: fail (misleading) 2026-10-18 — The first sentence incorrectly states that a coefficient is divided out; the equation e^{2x}=3 has no coefficient to remove. The correct step is simply taking the natural logarithm of both sides.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution only verifies that the proposed answer is a root, but does not demonstrate that it is the unique solution. Solving an equation requires showing equivalence or uniqueness, not just checking a candidate.gpt-oss:20b: fail (misleading) 2026-10-16 — The first sentence incorrectly states “divide out the coefficient,” which is unnecessary and misleading for the equation e^{2x}=3; the correct step is simply taking the natural logarithm of both sides.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the steps to isolate x and provides the correct 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.