If log 27 (to base 10) is 1.431, then what is the value of log 9 (to the same base, correct to three decimal places)?

Introduction / Context:
This question checks understanding of logarithm laws, especially how to relate logs of numbers that are powers of the same base. We are given the common logarithm of 27 and are asked to find the logarithm of 9. Both numbers are powers of 3, so we can express them in terms of log 3 and then calculate log 9 from the given numerical value of log 27.

Given Data / Assumptions:

• log 27 (base 10) = 1.431.
• We need to find log 9 (base 10).
• All logarithms are base 10 unless otherwise specified.
• 27 and 9 can be expressed as powers of 3: 27 = 3³ and 9 = 3².

Concept / Approach:
Use the rule log aⁿ = n log a. First express 27 and 9 in terms of 3. From log 27 = log 3³ = 3 log 3, we can find log 3. Then use log 9 = log 3² = 2 log 3. Substituting the value of log 3 obtained from the given information yields the required numerical value of log 9.

Step-by-Step Solution:
Step 1: Write 27 as a power of 3: 27 = 3³. Step 2: Apply the power rule: log 27 = log(3³) = 3 log 3. Step 3: Given that log 27 = 1.431, set 3 log 3 = 1.431, so log 3 = 1.431 / 3. Step 4: Compute log 3: 1.431 / 3 = 0.477. Step 5: Now write 9 as 3² and use the power rule again: log 9 = log(3²) = 2 log 3. Step 6: Substitute log 3 = 0.477 to obtain log 9 = 2 × 0.477 = 0.954.

Verification / Alternative check:
We can quickly check the consistency of these logs. If log 3 ≈ 0.477, then log 27 = log 3³ = 3 × 0.477 = 1.431, which matches the given data. Similarly, 10^0.954 is between 10^0.9 and 10^1.0, that is between about 7.94 and 10, which is reasonable since 9 lies in this range. This supports that 0.954 is the correct value for log 9.

Why Other Options Are Wrong:
Options 0.754, 0.854, and 0.654 do not satisfy the relationship between log 27 and log 9 when tested through log 3. Option 1.054 is larger than 1, which would correspond to a number greater than 10, not 9. Only 0.954 preserves the exact relationship log 27 = 1.431 and log 9 = 2 log 3 based on that value.

Common Pitfalls:
Some learners mistakenly divide the given log 27 by 2 instead of 3 or add or subtract arbitrary values. Others forget that the same logarithm base must be used consistently. Confusing 27 and 9 or misapplying the power rule is also common. Carefully expressing each number as a power of 3 and applying log rules avoids these errors.

Final Answer:
The value of log 9 (base 10) is 0.954 (approximately).