Find the domain `f(x)=sqrt((log_(0.3)|x-2|)/(|x|))`.

Explanation

To find the domain of the function \( f(x) = \sqrt{\frac{\log_{0.3} |x-2|}{|x|}} \), we need to ensure that the expression inside the square root is non-negative and that the denominator is not zero. Let's break this down step by step.

### Step 1: Ensure the denominator is not zero
The denominator is \( |x| \). For the function to be defined, we need:
\[ |x| > 0 \implies x \neq 0 \]

### Step 2: Ensure the expression inside the square root is non-negative
We need to ensure that:
\[ \frac{\log_{0.3} |x-2|}{|x|} \geq 0 \]
This implies:
\[ \log_{0.3} |x-2| \geq 0 \]
Since the base of the logarithm (0.3) is less than 1, the logarithm is non-negative when its argument is less than or equal to 1:
\[ |x-2| \leq 1 \]

### Step 3: Solve the inequality \( |x-2| \leq 1 \)
This inequality can be rewritten as:
\[ -1 \leq x-2 \leq 1 \]
Adding 2 to all parts of the inequality gives:
\[ 1 \leq x \leq 3 \]

### Step 4: Combine the conditions
From Step 1, we have \( x \neq 0 \). The interval we found in Step 3 is \( [1, 3] \). Since \( 0 \) is not in this interval, we do not need to exclude it from our domain.

### Step 5: Exclude the point where \( |x-2| = 0 \)
We also need to ensure that \( |x-2| \) does not equal zero, as this would make the logarithm undefined. Thus, we exclude \( x = 2 \).

### Final Domain
The domain of \( f(x) \) is:
\[ [1, 3] \setminus \{2\]
This can be expressed as:
\[ [1, 2) \cup (2, 3] \]

### Summary of the Domain
The domain of the function \( f(x) = \sqrt{\frac{\log_{0.3} |x-2|}{|x|}} \) is:
\[ [1, 2) \cup (2, 3] \]