The domain of the function
`f(x)=log_(e){log_(|sinx|)(x^(2)-8x+23)-(3)/(log_(2)|sinx|)}`
contains which of the following interval (s) ?

Options

A. `(3, pi)`
B. `(pi,(3pi)/(2))`
C. `((3pi)/(2),5)`
D. None of these

Explanation

To determine the domain of the function

\[ f(x) = \log_e \left( \log_{|\sin x|} (x^2 - 8x + 23) - \frac{3}{\log_2 |\sin x|} \right), \]

we need to ensure that the expression inside the logarithm is positive. This requires two conditions to be satisfied:

1. The argument of the logarithm must be greater than zero.
2. The base of the logarithm must be valid (i.e., it must be positive and not equal to 1).

### Step 1: Analyze the inner logarithm

The inner logarithm is

\[ \log_{|\sin x|} (x^2 - 8x + 23). \]

For this logarithm to be defined, we need:

- \( |\sin x| > 0 \) (which means \( \sin x \neq 0 \))
- \( x^2 - 8x + 23 > 0 \)

### Step 2: Solve the quadratic inequality

The quadratic expression \( x^2 - 8x + 23 \) can be analyzed using the discriminant:

\[ D = b^2 - 4ac = (-8)^2 - 4 \cdot 1 \cdot 23 = 64 - 92 = -28. \]

Since the discriminant is negative, the quadratic has no real roots and is always positive. Therefore, \( x^2 - 8x + 23 > 0 \) for all \( x \).

### Step 3: Analyze the base of the logarithm

Next, we need to ensure that \( |\sin x| \) is a valid base for the logarithm:

- \( |\sin x| > 0 \) implies \( \sin x \neq 0 \).
- \( |\sin x| \neq 1 \) implies \( \sin x \neq 1 \) and \( \sin x \neq -1 \).

### Step 4: Identify intervals where \( \sin x \neq 0 \)

The sine function is zero at integer multiples of \( \pi \):

\[ x = n\pi, \quad n \in \mathbb{Z}. \]

### Step 5: Identify intervals where \( \sin x \neq 1 \) and \( \sin x \neq -1 \)

The sine function equals 1 at \( x = \frac{\pi}{2} + 2k\pi \) and equals -1 at \( x = \frac{3\pi}{2} + 2k\pi \).

### Step 6: Combine the conditions

The function \( f(x) \) will be defined in intervals where \( \sin x \) is neither 0 nor ±1. Therefore, we exclude the points:

- \( n\pi \) (for \( n \in \mathbb{Z} \))
- \( \frac{\pi}{2} + 2k\pi \) (for \( k \in \mathbb{Z} \))
- \( \frac{3\pi}{2} + 2k\pi \) (for \( k \in \mathbb{Z} \))

### Step 7: Determine the domain intervals

From the analysis, we can conclude that the domain of \( f(x) \) excludes the points where \( \sin x \) is 0 or ±1. Thus, we can express the domain in intervals.

The intervals where \( f(x) \) is defined can be written as:

\[ (0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi) \cup ( \pi, \frac{3\pi}{2}) \cup (\frac{3\pi}{2}, 2\pi) \cup \ldots \]

### Final Domain

The domain of the function \( f(x) \) contains intervals excluding the points where \( \sin x = 0 \) or ±1.