Evaluate (log(a^n)) / (log(a^{bn})) and express the result in simplest symbolic form (a > 0, a ≠ 1; b, n ≠ 0).

Introduction / Context:
Simplify a ratio of logarithms with the same base and power arguments. Using log rules, powers can be pulled down as multipliers. Many expressions collapse cleanly when the same base and variable powers are present in numerator and denominator.

Given Data / Assumptions:

• a > 0, a ≠ 1 (log base conditions).
• b ≠ 0, n ≠ 0.
• Expression: (log(a^n)) / (log(a^{bn})).

Concept / Approach:
Use log power rule: log(a^k) = k * log(a) for any admissible base (here the unspecified common base cancels). Then simplify the scalar ratio.

Step-by-Step Solution:

Verification / Alternative check:
Pick concrete values, e.g., a = 10, n = 2, b = 3. Then (log(10^2))/(log(10^6)) = 2/6 = 1/3 = 1/b ✓.

Why Other Options Are Wrong:
They introduce extraneous terms or swap bases; the ratio purely cancels to 1/b by the power rule and common factor log(a).

Common Pitfalls:
Confusing log base with argument; forgetting both numerator and denominator share the same multiplicative factor log(a).

Final Answer:
1 / b