Given log 10(90) = 1.9542, find log 10(3). Use the relationships among 3, 9, and 90.
Aptitude
Logarithm
Difficulty: Easy
Choose an option
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A0.9771
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B0.6514
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C0.4771
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D0.3181
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ENone of these
Answer
Correct Answer: 0.4771
Explanation
Introduction / Context:This problem uses the factorization of 90 and standard log identities to isolate log 3.
Given Data / Assumptions:
- log 10(90) = 1.9542
- 90 = 9 × 10 = 3^2 × 10
Concept / Approach:Use log(AB) = log A + log B and power rules to relate log 90 to log 3.
Step-by-Step Solution:
log 90 = log(3^2 × 10) = 2·log 3 + log 101.9542 = 2·log 3 + 12·log 3 = 0.9542 ⇒ log 3 = 0.9542 / 2 = 0.4771Verification / Alternative check:Since 3 ≈ 10^0.4771, then 3^2 × 10 ≈ 10^0.9542 × 10 = 10^1.9542 ≈ 90, consistent with the given number.
Why Other Options Are Wrong:Other decimals do not satisfy 2·log 3 + 1 = 1.9542 when substituted; only 0.4771 does.
Common Pitfalls:Forgetting that log 10 = 1 or failing to divide by 2 after isolating 2·log 3 are common mistakes.
Final Answer:0.4771