The domain 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)/(2))^(pi), (pi)/(4))`
D. `(((pi)/(2))^(pi), oo)`

Explanation

To find the domain of the function \( f(x) = \log_{(3/2)} \log_{(1/2)} \log_{\pi} \log_{(\pi/4)} x \), we need to ensure that each logarithmic expression is defined and positive. Let's break this down step by step.

### Step 1: Determine the innermost logarithm
The innermost function is \( \log_{(\pi/4)} x \). For this logarithm to be defined, we need:
\[ x > 0 \]
Additionally, for \( \log_{(\pi/4)} x \) to be positive, we require:
\[ x > \frac{\pi}{4} \]

### Step 2: Move to the next logarithm
Next, we consider \( \log_{\pi} \log_{(\pi/4)} x \). For this logarithm to be defined and positive, we need:
\[ \log_{(\pi/4)} x > 0 \]
This implies:
\[ x > \frac{\pi}{4} \]
Since \( \log_{(\pi/4)} x \) is already positive when \( x > \frac{\pi}{4} \), we can proceed to the next logarithm.

### Step 3: Consider the logarithm with base \( \frac{1}{2} \)
Next, we analyze \( \log_{(1/2)} \log_{\pi} \log_{(\pi/4)} x \). For this logarithm to be defined and positive, we need:
\[ \log_{\pi} \log_{(\pi/4)} x < 1 \]
This means:
\[ \log_{(\pi/4)} x < \pi \]
We can rewrite this inequality using the property of logarithms:
\[ x < \left(\frac{\pi}{4}\right)^{\pi} \]

### Step 4: The outermost logarithm
Finally, we consider \( \log_{(3/2)} \log_{(1/2)} \log_{\pi} \log_{(\pi/4)} x \). For this logarithm to be defined and positive, we need:
\[ \log_{(1/2)} \log_{\pi} \log_{(\pi/4)} x > 0 \]
This implies:
\[ \log_{\pi} \log_{(\pi/4)} x < \frac{1}{2} \]
This means:
\[ \log_{(\pi/4)} x < \pi^{1/2} \]
Rearranging gives:
\[ x < \left(\frac{\pi}{4}\right)^{\sqrt{\pi}} \]

### Step 5: Combining all conditions
Now we combine all the conditions we derived:
1. \( x > \frac{\pi}{4} \)
2. \( x < \left(\frac{\pi}{4}\right)^{\pi} \)
3. \( x < \left(\frac{\pi}{4}\right)^{\sqrt{\pi}} \)

Thus, the domain of the function \( f(x) \) can be expressed as:
\[ \frac{\pi}{4} < x < \min\left(\left(\frac{\pi}{4}\right)^{\pi}, \left(\frac{\pi}{4}\right)^{\sqrt{\pi}}\right) \]

### Final Domain
The domain of the function \( f(x) \) is:
\[ \left(\frac{\pi}{4}, \left(\frac{\pi}{4}\right)^{\min(\pi, \sqrt{\pi})}\right) \]