Question:medium

The number of distinct integer values of n satisfying \(4−\log\frac{2n}{3}−\log4n\lt0\), is

Updated On: Jan 15, 2026
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Correct Answer: 47

Solution and Explanation

The given inequality is: \(\frac{4 - \log_2 n}{3 - \log_4 n} < 0\)
Analysis:
1. \(\log_2 n = 4\) implies \(n = 2^4 = 16\).
2. \(\log_4 n = 3\) implies \(n = 4^3 = 64\).
For the fraction to be negative, the numerator and denominator must have opposite signs.
- Numerator (\(4 - \log_2 n\)) is positive for \(n < 16\) and negative for \(n > 16\).
- Denominator (\(3 - \log_4 n\)) is positive for \(n < 64\) and negative for \(n > 64\).
To achieve a negative fraction, we require (Numerator<0 AND Denominator>0) OR (Numerator>0 AND Denominator<0).
This occurs when \(16 < n < 64\).
The number of distinct integers in this range is \(64 - 16 - 1 = 47\).
Therefore, the answer is 47.
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