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If x is a real number, then $$\sqrt{\log_{e}{\dfrac{4x - x^2}{3}}}$$ is a real number if and only if
$$\sqrt{\log_{e}{\dfrac{4x - x^2}{3}}}$$ will be real if $$\log_e\ \dfrac{\ 4x-x^2}{3}\ \ge\ 0$$
$$\dfrac{\ 4x-x^2}{3}\ \ge\ 1$$
$$\ 4x-x^2-3\ \ge\ 0$$
$$\ x^2-4x+3\ \le\ 0$$
$$1\le\ x\le\ 3$$
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