Which of the following is/are true?
(you may use `f(x)=ln((lnx))/(lnx)`

Options

A. `(In2.1)^(In2.2)gt(In2.2)^(In2.1)`
B. `(In4)^(In5)gt(In5)^(In4)`
C. `(In30)^(In31)gt(In31)^(In30)`
D. `(In28)^(30)lt(In30)^(In28)`

Answer

To solve the problem, we need to analyze the function \( f(x) = \frac{\ln(\ln x)}{\ln x} \) and determine the intervals where it is increasing or decreasing. We will differentiate \( f(x) \) and analyze the sign of the derivative.

### Step 1: Differentiate \( f(x) \)

We start with the function:
\[ f(x) = \frac{\ln(\ln x)}{\ln x} \]

Using the quotient rule, the derivative \( f'(x) \) is given by:
\[ f'(x) = \frac{(\ln x)(\frac{1}{\ln x} \cdot \frac{1}{x}) - \ln(\ln x)(\frac{1}{x})}{(\ln x)^2} \]
This simplifies to:
\[ f'(x) = \frac{1 - \frac{\ln(\ln x)}{x}}{(\ln x)^2} \]

### Step 2: Analyze the sign of \( f'(x) \)

Now, we need to determine when \( f'(x) > 0 \) or \( f'(x) < 0 \):
\[ f'(x) > 0 \implies 1 - \frac{\ln(\ln x)}{x} > 0 \implies \ln(\ln x) < x \]

### Step 3: Find critical points

To find the critical points, we set \( f'(x) = 0 \):
\[ 1 - \frac{\ln(\ln x)}{x} = 0 \implies \ln(\ln x) = x \]

This equation is complex, but we can analyze it numerically or graphically to find that it holds true for \( x = e^e \).

### Step 4: Determine intervals of increase and decrease

1. **For \( x < e^e \)**:
- \( f'(x) > 0 \) (function is increasing)

2. **For \( x > e^e \)**:
- \( f'(x) < 0 \) (function is decreasing)

### Step 5: Conclusion about the function

From the analysis above, we conclude:
- The function \( f(x) \) is increasing on the interval \( (1, e^e) \) and decreasing on the interval \( (e^e, \infty) \).

### Step 6: Evaluate the options

Now we need to evaluate the options given in the problem. We need to check:
\[ \ln(4^{\ln 5}) < \frac{\ln 5}{\ln 4} \]

Using properties of logarithms, we can rewrite:
\[ \ln(4^{\ln 5}) = \ln 5 \cdot \ln 4 \]

Thus, we need to check if:
\[ \ln 5 \cdot \ln 4 < \frac{\ln 5}{\ln 4} \]

This inequality holds true when \( \ln 4 \) is greater than 1, which it is. Therefore, the statement is true.

### Final Answer

Thus, the correct option is:
- Option B: \( \ln(4^{\ln 5}) < \frac{\ln 5}{\ln 4} \)

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