The function $f(x) = \log_e(\sin x), x \in (0, \pi)$ is

(A) strictly increasing on $\left(0, \frac{\pi}{2}\right)$

(B) strictly decreasing on $\left(0, \frac{\pi}{2}\right)$

(C) strictly increasing on $\left(\frac{\pi}{2}, \pi\right)$

(D) strictly decreasing on $\left(\frac{\pi}{2}, \pi\right)$

(E) strictly increasing on $(0, \pi)$

Choose the correct answer from the options given below:

To determine where $f(x) = \log_e(\sin x)$ is increasing or decreasing, find the derivative.

Using the chain rule:

$f(x) = \log_e(\sin x)$

$f'(x) = \frac{1}{\sin x} \times \cos x$

$f'(x) = \frac{\cos x}{\sin x}$

$f'(x) = \cot x$

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For interval $(0, \frac{\pi}{2})$:

In this interval, $\sin x > 0$ and $\cos x > 0$.

Therefore $\cot x = \frac{\cos x}{\sin x} > 0$

Since $f'(x) > 0$, the function is strictly increasing on $(0, \frac{\pi}{2})$.

Statement (A) is correct.

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For interval $(\frac{\pi}{2}, \pi)$:

In this interval, $\sin x > 0$ and $\cos x < 0$.

Therefore $\cot x = \frac{\cos x}{\sin x} < 0$

Since $f'(x) < 0$, the function is strictly decreasing on $(\frac{\pi}{2}, \pi)$.

Statement (D) is correct.

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Statements (B), (C), and (E) are incorrect based on the analysis above.

The correct answer is (A) and (D).