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Exponential and logarithmic equations

Problem 13.106 · easy

Solve \( \displaystyle \ln_{2}\left(4 x + 5\right) = 0 \) exactly.

Rewrite in exponential form: 4*x + 5 = 2^0.
\[ \left. \frac{\ln{\left(4 x + 5 \right)}}{\ln{\left(2 \right)}} \right|_{\substack{ x=-1 }} = 0 \]
x = -1 satisfies the equation.✓ Proved

Answer \( x = -1 \)

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 (error) — The solution only verifies that x = -1 is a solution but does not demonstrate that it is the unique solution. Solving an equation requires showing that no other values satisfy it, typically by deriving the solution from the equation rather than just checking a candidate.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution only verifies that x = -1 is a solution but does not demonstrate that it is the unique solution. Solving an equation requires showing that no other values satisfy it, typically by deriving the solution from the equation rather than just checking a candidate.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution only verifies that x = -1 is a solution but fails to demonstrate that it is the unique solution. Solving an equation requires showing equivalence or uniqueness, not just checking a candidate.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the definition of logarithms to convert the equation to exponential form and solves for x. The verification step confirms the solution is valid.
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.