If log 27 (to base 10) is 1.431, then what is the value of log 9 (to base 10)?

Introduction / Context:
This problem checks your familiarity with the laws of logarithms and the ability to express logs of related numbers in terms of each other. You are given log 27 and asked to find log 9, both with the same base, by recognising that 27 and 9 are powers of 3.

Given Data / Assumptions:

• log 27 = 1.431 (base 10).
• We need log 9 to the same base.
• 27 = 3³ and 9 = 3².

Concept / Approach:
The central idea is to rewrite each number as a power of the same base, here base 3, and then use the rule log(a^k) = k log a. Since 27 and 9 share the same prime base, 3, we can solve for log 3 first using the given information and then multiply it by 2 to find log 9.

Step-by-Step Solution:
Write 27 as 3³. Then log 27 = log(3³) = 3 log 3. We are told log 27 = 1.431, so 3 log 3 = 1.431. Hence log 3 = 1.431 / 3. Compute the division: 1.431 / 3 = 0.477. Now express 9 as 3². Then log 9 = log(3²) = 2 log 3. Substitute log 3: log 9 = 2 × 0.477 = 0.954.

Verification / Alternative check:
We know approximately that log 3 is around 0.4771 in standard tables. Doubling that gives about 0.9542, which agrees closely with the calculated 0.954 and matches the given multiple-choice option. So the answer is consistent with known values.

Why Other Options Are Wrong:
0.754 and 0.654: These values are too small and would correspond to squares of numbers less than 3.
0.854: This is also smaller than the correct value. Since 9 is greater than 7 and less than 10, its logarithm should be between log 7 and 1, which suggests a value closer to 0.95 than to 0.85.

Common Pitfalls:
A common mistake is to divide 1.431 by 2 instead of 3 when solving for log 3, or to forget that log 9 equals 2 log 3, not log 3² treated in some other way. Another error is attempting to approximate logs of 9 directly without using the relationship with log 3, which is slower and less accurate.

Final Answer:
The value of log 9 is 0.954.