Given 10^0.3010 = 2, evaluate log base 0.125 of 125, i.e., log_{0.125}(125). Show the change-of-base steps.

Introduction / Context:
We compute a logarithm with a base less than 1 using change-of-base. Given log 10(2), we can find log 10(5) and then express the result exactly as a rational combination of these constants.

Given Data / Assumptions:

• 10^0.3010 = 2 ⇒ log 10(2) = 0.3010
• We need log_{0.125}(125)

Concept / Approach:
Rewrite bases as powers: 0.125 = 1/8 = 2^(−3), and 125 = 5^3. Then log_{2^(−3)}(5^3) = (3 ln 5)/(−3 ln 2) = − ln 5 / ln 2. Convert to base 10 using given log values.

Step-by-Step Solution:

Verification / Alternative check:
Since the base 0.125 < 1 and argument 125 > 1, the logarithm should be negative; the sign matches. The exact fraction −699/301 is the simplified ratio of common logs.

Why Other Options Are Wrong:
Positive 699/301 contradicts the expected sign; −1 and −2 are rough but incorrect; only −699/301 matches the exact change-of-base computation.

Common Pitfalls:
Forgetting that a base less than 1 flips the sign of logs relative to the same argument, or failing to compute log 5 = 1 − log 2 correctly.

Final Answer:
- 699 / 301