Home›Calculus 1›Exponential and logarithmic equations›Problem 13.387

Exponential and logarithmic equations

Problem 13.387 · easy

Solve \( \displaystyle 2^{2 x + 3} = 4 \) exactly.

Write 4 as 2^2 and set the exponents equal.
Reviewed
\[ \left. 2^{2 x + 3} - 4 \right|_{\substack{ x=- \frac{1}{2} }} = 0 \]
x = -1/2 satisfies the equation.✓ Proved

Answer \( x = - \frac{1}{2} \)

Lines: 1 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies that the bases are equal and sets the exponents equal, which is a valid method for solving exponential equations with the same base. The final answer is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies that the bases are equal and sets the exponents equal, which is a valid method for solving exponential equations with the same base. The final answer is correct.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies that the bases are equal and sets the exponents equal, which is the standard method for solving such exponential equations. The verification step confirms the result.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution skips the algebraic step of equating the exponents (2x + 3 = 2) and solving for x. It only verifies the answer, which does not demonstrate how to solve the equation.
gpt-oss:20b: pass 2026-10-18

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 2026-10-19 with SymPy 1.14.0.