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Exponential and logarithmic equations

Problem 13.109 · easy

Solve \( \displaystyle e^{2 x} = 5 \) exactly.

Divide out the coefficient, then take ln of both sides.
\[ \left. e^{2 x} - 5 \right|_{\substack{ x=\frac{\ln{\left(5 \right)}}{2} }} = 0 \]
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 claims to 'divide out the coefficient,' but there is no coefficient to divide out in the equation e^{2x} = 5. This describes an incorrect algebraic step, even though the final answer is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (misleading) 2026-10-17 — The solution claims to 'divide out the coefficient,' but there is no coefficient to divide out in the equation e^{2x} = 5. This describes an incorrect algebraic step, even though the final answer is correct.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (misleading) 2026-10-14 — The solution claims to 'divide out the coefficient,' but there is no coefficient to divide; the correct step is to take the natural logarithm of both sides. This misstates the algebraic procedure.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the algebraic steps to isolate x and verifies the result. The sentence accurately describes the method used.
gpt-oss:20b: fail (misleading) 2026-10-14 — The sentence claims to divide out a coefficient, but the equation e^{2x}=5 has no coefficient to divide. The correct step is to take the natural logarithm of both sides directly. The statement misleads a student into thinking an unnecessary division step is needed.

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.