Compute log₉(81) − log₄(32). Express each exactly via prime powers and subtract.
Aptitude
Logarithm
Difficulty: Easy
Choose an option
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A1 / 2
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B- 3 / 2
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C- 1 / 2
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D2
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ENone of these
Answer
Correct Answer: - 1 / 2
Explanation
Introduction / Context:By expressing each base and argument as a power of a common prime, we can evaluate the logarithms exactly and then subtract.
Given Data / Assumptions:
- 9 = 3^2, 81 = 3^4
- 4 = 2^2, 32 = 2^5
Concept / Approach:Use log_{p^m}(p^n) = n/m to get exact rational values.
Step-by-Step Solution:
log₉(81) = log_{3^2}(3^4) = 4/2 = 2log₄(32) = log_{2^2}(2^5) = 5/2 = 2.5Difference: 2 − 5/2 = −1/2Verification / Alternative check:Change of base to ln confirms the same rational numbers and difference.
Why Other Options Are Wrong:2 is the first term, not the difference; 1/2 has the wrong sign; −3/2 would require 2 − 3.5, which is not the case.
Common Pitfalls:Mixing up which number is base and which is argument or reversing the subtraction order changes the sign.
Final Answer:- 1 / 2