2 1) \( \log _{3}(5-4 x)<\log _{3}(x-1) \)

To solve the inequality \( \log _{3}(5-4x) < \log _{3}(x-1) \), we start by using the properties of logarithms, which allow us to deduce that \( 5-4x < x-1 \) if both logarithmic terms are defined and positive. This leads us to the inequality \( 5 + 1 < x + 4x \), simplifying to \( 6 < 5x \) or \( x > \frac{6}{5} \). Next, we need to ensure the domains of the logarithms are valid, which means: 1. \( 5 - 4x > 0 \) leading to \( x < \frac{5}{4} \) 2. \( x - 1 > 0 \) leading to \( x > 1 \) By combining these results, we find that the solution set is \( 1 < x < \frac{5}{4} \) and additionally \( x > \frac{6}{5} \), meaning the valid solution range becomes \( \frac{6}{5} < x < \frac{5}{4} \). Thus, the complete solution to the inequality is \( \frac{6}{5} < x < \frac{5}{4} \).