Evaluate log_{1/3}(81). (Assume real-valued logarithms where the base is positive and not equal to 1.)

Introduction / Context:The expression asks for the exponent k such that (1/3)^k = 81. Recognizing 81 as a power of 3 allows a quick evaluation by matching bases and exponents.

Given Data / Assumptions:

• Compute k where (1/3)^k = 81.
• 81 = 3^4.

Concept / Approach:Use base transformation: (1/3)^k = 3^{−k}. Set 3^{−k} equal to 3^4 and equate exponents to solve for k.

Step-by-Step Solution:

Verification / Alternative check:(1/3)^{−4} = 3^4 = 81 ✓.

Why Other Options Are Wrong:

• 4 changes the sign (would satisfy 3^k = 81, not (1/3)^k = 81).
• −27 and 127 are unrelated large-magnitude distractors.

Common Pitfalls:Forgetting (1/3)^k = 3^{−k}, leading to incorrect sign on the exponent.

Final Answer:−4