Evaluate the difference: log₄₉(16807) − log₉(27). Convert to prime powers first to obtain an exact rational result.
Aptitude
Logarithm
Difficulty: Easy
Choose an option
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A0
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B1
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C3 / 2
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D- 1
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ENone of these
Answer
Correct Answer: 1
Explanation
Introduction / Context:We compute two exact logarithms by expressing both bases and arguments as powers of a common prime and then subtract.
Given Data / Assumptions:
- 49 = 7^2, 16807 = 7^5
- 9 = 3^2, 27 = 3^3
Concept / Approach:Use log_{p^m}(p^n) = n/m. Then subtract the two rational numbers obtained.
Step-by-Step Solution:
log₄₉(16807) = log_{7^2}(7^5) = 5/2log₉(27) = log_{3^2}(3^3) = 3/2Difference: 5/2 − 3/2 = 1Verification / Alternative check:Change of base to natural logs gives ln(7^5)/ln(7^2) − ln(3^3)/ln(3^2) = (5/2) − (3/2) as above.
Why Other Options Are Wrong:0 or −1 would require equal or reversed fractions; 3/2 is one of the individual logs, not the difference.
Common Pitfalls:Mistaking 16807 as 7^6 or misreading 49/9 bases leads to wrong fractions. Confirm prime-power factorizations first.
Final Answer:1