Logarithmic Inequality Integer Solutions

Rewrite $\log_4 n = \frac{\log_2 n}{2}$. Let $x = \log_2 n$. The inequality becomes: $\frac{4 - x}{3 - \frac{x}{2}} < 0 \implies \frac{4 - x}{\frac{6 - x}{2}} < 0 \implies \frac{x - 4}{x - 6} < 0$. This inequality holds when $4 < x < 6$. Substituting $x = \log_2 n$: $4 < \log_2 n < 6 \implies 2^4 < n < 2^6 \implies 16 < n < 64$. Integer values of $n$ are $17, 18, \dots, 63$. Number of integers = $63 - 17 + 1 = 47$.