Home›Calculus 1›Domain of a function›Problem 13.205

Domain of a function

Problem 13.205 · easy

Find the domain of \( \displaystyle f(x) = \ln{\left(x - 3 \right)} + 1 \). Give it in interval notation.

\[ \left. x - 3 \right|_{\substack{ x=3 }} = 0 \]
At x = 3 the log's argument is 0 (it must be > 0).✓ Proved
Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.
Reviewed

Answer \( (3, \infty) \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	f evaluated at 321 points in [−10, 10] is real exactly where the claimed domain says

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the condition for the logarithm's argument to be positive and states the general rule for finding domains. The final answer (3, infinity) is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the condition for the logarithm's argument to be positive and states the general rule for finding domains. The final answer (3, infinity) is correct.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the domain constraint for the natural logarithm (argument must be positive) and provides the correct interval notation. The general statement in line 2 is accurate and applicable.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the condition for the logarithm's argument to be positive and provides the correct domain.
gpt-oss:20b: pass 2026-10-15

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/domain_range, checked 2026-10-18 with SymPy 1.14.0.