The domain of definition of the function `f(x) = log_(3//2) log_(1//2) log_(pi) log_(pi //4) x` is

Options

A. `(0, oo)`
B. `(0, ((pi)/(4))^(pi))`
C. `(((pi)/(4))^(pi),(pi)/(4))`
D. `(((pi)/(2))^(pi),oo)`

Answer

To find the domain of the function \( f(x) = \log_{\frac{3}{2}} \left( \log_{\frac{1}{2}} \left( \log_{\pi} \left( \log_{\frac{\pi}{4}} x \right) \right) \right) \), we need to ensure that all logarithmic expressions are defined and valid.

### Step 1: Determine the innermost logarithm
The innermost logarithm is \( \log_{\frac{\pi}{4}} x \). For this logarithm to be defined, we need:
1. \( x > 0 \) (since the argument of a logarithm must be positive).
2. \( \frac{\pi}{4} > 0 \) (which is always true).

### Step 2: Set the condition for the innermost logarithm
Next, we need \( \log_{\frac{\pi}{4}} x > 0 \). This implies:
\[ x > \frac{\pi}{4}^1 \quad \text{(since } \frac{\pi}{4} < 1 \text{)} \]
So, we have:
\[ x > \frac{\pi}{4} \]

### Step 3: Move to the next logarithm
Now, we consider \( \log_{\pi} \left( \log_{\frac{\pi}{4}} x \right) \). For this logarithm to be defined, we need:
1. \( \log_{\frac{\pi}{4}} x > 0 \), which we already established means \( x > \frac{\pi}{4} \).
2. \( \pi > 0 \) (which is always true).

### Step 4: Set the condition for the second logarithm
Now we need \( \log_{\pi} \left( \log_{\frac{\pi}{4}} x \right) > 0 \). This implies:
\[ \log_{\frac{\pi}{4}} x > 1 \quad \text{(since } \pi > 1 \text{)} \]
This leads to:
\[ x > \left( \frac{\pi}{4} \right)^1 = \frac{\pi}{4} \]

### Step 5: Move to the next logarithm
Next, we consider \( \log_{\frac{1}{2}} \left( \log_{\pi} \left( \log_{\frac{\pi}{4}} x \right) \right) \). For this logarithm to be defined, we need:
1. \( \log_{\pi} \left( \log_{\frac{\pi}{4}} x \right) > 0 \), which we established means \( \log_{\frac{\pi}{4}} x > 1 \).
2. \( \frac{1}{2} > 0 \) (which is always true).

### Step 6: Set the condition for the third logarithm
Now we need \( \log_{\frac{1}{2}} \left( \log_{\pi} \left( \log_{\frac{\pi}{4}} x \right) \right) > 0 \). This implies:
\[ \log_{\pi} \left( \log_{\frac{\pi}{4}} x \right) < 1 \quad \text{(since } \frac{1}{2} < 1 \text{)} \]
This leads to:
\[ \log_{\frac{\pi}{4}} x < \pi \quad \text{(since } \pi > 1 \text{)} \]
Thus:
\[ x < \left( \frac{\pi}{4} \right)^{\pi} \]

### Step 7: Combine the conditions
We now have two conditions:
1. \( x > \frac{\pi}{4} \)
2. \( x < \left( \frac{\pi}{4} \right)^{\pi} \)

### Conclusion
Thus, the domain of the function \( f(x) \) is:
\[ x \in \left( \frac{\pi}{4}, \left( \frac{\pi}{4} \right)^{\pi} \right) \]

### Final Answer
The domain of the function \( f(x) \) is \( \left( \frac{\pi}{4}, \left( \frac{\pi}{4} \right)^{\pi} \right) \).

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