In which of the following interval, the function $f(x) =\frac{x}{\log x}$ is decreasing?

$(0, e) - \{1\}$

The correct answer is Option (3) → $(0, e) - \{1\}$

Given function: $f(x) = \frac{x}{\log x}$

Derivative: $f'(x) = \frac{\log x - 1}{(\log x)^2}$

Function decreases when $f'(x) < 0$: $\log x - 1 < 0 \Rightarrow \log x < 1 \Rightarrow x < e$

Domain: $x>0$ and $x \neq 1$ (since $\log 1 = 0$)

Hence, $f(x)$ is decreasing on $(0, e) - \{1\}$