Log3(5-4x)

To solve the inequality \( \log_3(5 - 4x) < \log_3(x - 1) \), we can start by using the properties of logarithms. Specifically, since the logarithmic function is increasing when the base is greater than 1, we can simplify the inequality as follows: \[ 5 - 4x < x - 1 \] Next, let's solve this inequality step by step: 1. Rearrange the terms: \[ 5 + 1 < x + 4x \] This simplifies to: \[ 6 < 5x \] 2. Divide both sides by 5: \[ \frac{6}{5} < x \] Or: \[ x > \frac{6}{5} \] Now, we must also consider the conditions under which the logarithms are defined. The arguments of the logarithms must be positive: 1. From \( \log_3(5 - 4x) \), we need: \[ 5 - 4x > 0 \implies 4x < 5 \implies x < \frac{5}{4} \] 2. From \( \log_3(x - 1) \), we need: \[ x - 1 > 0 \implies x > 1 \] Now we have three inequalities to satisfy: 1. \( x > \frac{6}{5} \) 2. \( x < \frac{5}{4} = 1.25 \) 3. \( x > 1 \) Considering the inequalities \( x > \frac{6}{5} \) and \( x < \frac{5}{4} \), the solution will be between the two: \[ \frac{6}{5} < x < \frac{5}{4} \] In interval notation, the solution can be expressed as: \[ \left( \frac{6}{5}, \frac{5}{4} \right) \] So the final solution is: \[ x \in \left( \frac{6}{5}, \frac{5}{4} \right) \]