Find the interval of the monotonicity of the function f(x)=`log_(e)((log_(e)x)/(x))`

Explanation

To find the interval of monotonicity of the function \( f(x) = \ln\left(\frac{\ln x}{x}\right) \), we will follow these steps:

### Step 1: Determine the Domain of the Function
The function \( f(x) \) is defined when \( \frac{\ln x}{x} > 0 \).

- For \( x > 1 \), \( \ln x > 0 \) and \( x > 0 \), hence \( \frac{\ln x}{x} > 0 \).
- For \( 0 < x < 1 \), \( \ln x < 0 \) and \( x > 0 \), hence \( \frac{\ln x}{x} < 0 \).

Thus, the domain of \( f(x) \) is \( x \in (1, \infty) \).

### Step 2: Find the Derivative \( f'(x) \)
Using the quotient rule, we differentiate \( f(x) \):

\[ f'(x) = \frac{d}{dx} \left( \ln\left(\frac{\ln x}{x}\right) \right) = \frac{1}{\frac{\ln x}{x}} \cdot \frac{d}{dx}\left(\frac{\ln x}{x}\right) \]

Now, we need to differentiate \( \frac{\ln x}{x} \) using the quotient rule:

\[ \frac{d}{dx}\left(\frac{\ln x}{x}\right) = \frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x^2} = \frac{1 - \ln x}{x^2} \]

Thus, substituting back, we have:

\[ f'(x) = \frac{1}{\frac{\ln x}{x}} \cdot \frac{1 - \ln x}{x^2} = \frac{x(1 - \ln x)}{\ln x} \]

### Step 3: Analyze the Sign of \( f'(x) \)
To find the intervals of monotonicity, we need to determine where \( f'(x) > 0 \) and \( f'(x) < 0 \).

1. **Set \( f'(x) = 0 \):**
\[ x(1 - \ln x) = 0 \]
This gives \( 1 - \ln x = 0 \) or \( \ln x = 1 \), which implies \( x = e \).

2. **Test intervals around \( x = e \):**
- For \( x < e \): Choose \( x = 2 \)
\[ f'(2) = \frac{2(1 - \ln 2)}{\ln 2} > 0 \quad (\text{since } \ln 2 < 1) \]
- For \( x > e \): Choose \( x = 4 \)
\[ f'(4) = \frac{4(1 - \ln 4)}{\ln 4} < 0 \quad (\text{since } \ln 4 > 1) \]

### Step 4: Conclusion on Monotonicity
From the analysis:
- \( f'(x) > 0 \) for \( x \in (1, e) \) (function is increasing)
- \( f'(x) < 0 \) for \( x \in (e, \infty) \) (function is decreasing)

Thus, the intervals of monotonicity for the function \( f(x) \) are:
- Increasing on \( (1, e) \)
- Decreasing on \( (e, \infty) \)

### Final Answer
The function \( f(x) = \ln\left(\frac{\ln x}{x}\right) \) is increasing on the interval \( (1, e) \) and decreasing on the interval \( (e, \infty) \).

---