Let $f(x)= \log_e (\sin x), x ∈ (0,π)$, then which of the following statements is/are TRUE?

(A) f(x) is increasing on (0, π/2)
(B) f(x) is decreasing on (π/2, π)
(C) f(x) is increasing on (0, π)
(D) f(x) is decreasing on (0, π)

Choose the correct answer from the options given below:

(A) and (B) only

The correct answer is Option (2) → (A) and (B) only

Given

$f(x) = \log_e(\sin x), \quad x \in (0, \pi)$

Differentiate:

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

Now check the sign of $f'(x)$:

• For $x \in \left(0, \frac{\pi}{2}\right)$, $\cot x > 0$ $\Rightarrow f(x)$ is increasing.
• For $x \in \left(\frac{\pi}{2}, \pi\right)$, $\cot x < 0$ $\Rightarrow f(x)$ is decreasing.