If log_12 27 = a, find log_6 16 in terms of a.

Introduction / Context:
We are asked to rewrite one logarithm in terms of another, using change-of-base and prime factorization to express both in terms of ln 2 and ln 3. Setting t = ln 3 / ln 2 simplifies algebraic manipulation.

Given Data / Assumptions:

• a = log_12 27 = ln 27 / ln 12.
• We want log_6 16 = ln 16 / ln 6.
• Prime factorizations: 27 = 3^3, 12 = 3 * 2^2, 16 = 2^4, 6 = 2 * 3.

Concept / Approach:

• Let t = ln 3 / ln 2; then a = 3t / (t + 2).
• Compute log_6 16 = (4 ln 2) / (ln 2 + ln 3) = 4 / (1 + t).

Step-by-Step Solution:

Verification / Alternative check:
Choose numeric a via ln to test; substitution confirms the closed form 4(3 − a)/(3 + a).

Why Other Options Are Wrong:

• Other forms invert signs or reciprocals incorrectly.

Common Pitfalls:

• Mishandling algebra when isolating t from a = 3t/(t + 2).

Final Answer:
4(3 - a) / (3 + a)