Evaluate log base 0.125 of 64, i.e., log_{0.125}(64). Write both the power-of-2 reduction and the final numeric value.

Introduction / Context:
We evaluate a logarithm with base less than 1 by expressing base and argument as powers of a common number (2) and then using exponent division.

Given Data / Assumptions:

• Base: 0.125 = 1/8 = 2^(−3)
• Argument: 64 = 2^6

Concept / Approach:
Use log_{a^m}(a^n) = n/m. For a base less than 1 (negative exponent on 2), the result may be negative even with a > 1 argument.

Step-by-Step Solution:

Verification / Alternative check:
Check: (0.125)^(−2) = (1/8)^(−2) = 8^2 = 64, confirming the result.

Why Other Options Are Wrong:
2 has the wrong sign for a base less than 1; 0 or “can’t be determined” contradict the exact exponent check above.

Common Pitfalls:
Ignoring that a fractional base (<1) leads to negative log values for arguments >1 is a common source of sign errors.

Final Answer:
- 2