Question

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

Answer 1

Correct Answer: 47

The given inequality is: \(\frac{4 - \log_2 n}{3 - \log_4 n} < 0\)
Let's analyze this step by step:
1. We know that \(\log_2 n = 4\) when \(n = 2^4 = 16\). 2. Similarly, \(\log_4 n = 3\) when \(n = 4^3 = 64\).
Now, we want the fraction to be negative, which means the numerator and the denominator must have opposite signs. - For any value of n less than 16, the numerator is positive. - For any value of n greater than 16, the numerator is negative. - For any value of n less than 64, the denominator is positive. - For any value of n greater than 64, the denominator is negative. To make the fraction negative, we need either the numerator to be negative and the denominator to be positive, or vice versa. This condition is satisfied when n is between 16 and 64. So, the number of distinct integer values of n satisfying the inequality is the number of integers between 16 and 64, which is \(64 - 16 - 1 = 47\)(subtracting 1 because we don't want to count 16 and 64).
Hence, the answer is 47.