Domain for real logarithmic square root

For the square root of a logarithm to be real, the quantity inside the square root must be non-negative: $\log_e \left(\frac{4x-x^2}{3}\right) \ge 0$ $\frac{4x-x^2}{3} \ge e^0 = 1$ $4x - x^2 \ge 3$ $x^2 - 4x + 3 \le 0$ $(x - 1)(x - 3) \le 0 \implies 1 \le x \le 3$