Home›Calculus 1›Exponential and logarithmic equations›Problem 13.115

Exponential and logarithmic equations

Problem 13.115 · easy

Solve \( \displaystyle \ln_{10}\left(3 x - 4\right) = 0 \) exactly.

Rewrite in exponential form: 3*x - 4 = 10^0.
\[ \left. \frac{\ln{\left(3 x - 4 \right)}}{\ln{\left(10 \right)}} \right|_{\substack{ x=\frac{5}{3} }} = 0 \]
x = 5/3 satisfies the equation.✓ Proved

Answer \( x = \frac{5}{3} \)

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 (error) — The solution only verifies that x=5/3 satisfies the equation; it does not actually solve for x. The step of solving 3x-4=10^0 to obtain x=5/3 is missing.
qwen3.6:27b-mlx: pass — The solution correctly converts the logarithmic equation to exponential form and solves for x. The verification step confirms the result.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly converts the logarithmic equation to exponential form and solves for x. The verification step confirms the result.
gpt-oss:20b: fail (error) 2026-10-17 — The solution only verifies that x=5/3 satisfies the equation; it does not actually solve for x. The step of solving 3x-4=10^0 to obtain x=5/3 is missing.
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 result.
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 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 2026-10-17 with SymPy 1.14.0.