1. Derivative Calculation: The derivative of \( f(x) = \frac{\log x}{x} \) is computed as \( f'(x) = \frac{1 \cdot x - \log x \cdot 1}{x^2} = \frac{x - \log x}{x^2}. \)
2. Critical Points Determination: Critical points are found by setting \( f'(x) = 0 \), which yields \( x - \log x = 0 \quad \Rightarrow \quad x = \log x \). This equality is satisfied when \( x = e \), given the substitution \( x = e^k \).
3. Sign Analysis of \( f'(x) \): - For \( x \in (0, e) \), \( x - \log x>0 \), resulting in \( f'(x)>0 \), indicating that \( f(x) \) is increasing. - For \( x \in (e, \infty) \), \( x - \log x<0 \), resulting in \( f'(x)<0 \), indicating that \( f(x) \) is decreasing.
Monotonicity Intervals: \[ f(x) { is strictly increasing on } (0, e) { and strictly decreasing on } (e, \infty). \]