Find the domain of the function
`log_(x) log _(2) ((1)/(x - 1//2))`

Answer

To find the domain of the function \( f(x) = \log_{x} \left( \log_{2} \left( \frac{1}{x - \frac{1}{2}} \right) \right) \), we need to consider the restrictions imposed by the logarithmic functions involved.

### Step 1: Determine the conditions for the base of the logarithm
The base of the logarithm, \( x \), must satisfy the following conditions:
1. \( x > 0 \) (the base must be positive)
2. \( x \neq 1 \) (the base cannot be 1)

### Step 2: Analyze the inner logarithmic function
Next, we need to ensure that the argument of the inner logarithm, \( \log_{2} \left( \frac{1}{x - \frac{1}{2}} \right) \), is positive. This means:
\[ \frac{1}{x - \frac{1}{2}} > 0 \]
This inequality holds when:
\[ x - \frac{1}{2} > 0 \implies x > \frac{1}{2} \]

### Step 3: Set the argument of the inner logarithm greater than 1
For the logarithm \( \log_{2} \left( \frac{1}{x - \frac{1}{2}} \right) \) to be defined and positive, we need:
\[ \frac{1}{x - \frac{1}{2}} > 1 \]
This simplifies to:
\[ 1 > x - \frac{1}{2} \implies x < \frac{3}{2} \]

### Step 4: Combine the conditions
Now we have two conditions:
1. \( x > \frac{1}{2} \)
2. \( x < \frac{3}{2} \)

### Step 5: Exclude the point where the base is 1
Since \( x \neq 1 \), we must exclude this point from our solution.

### Final Domain
Combining all these conditions, the domain of the function is:
\[ \left( \frac{1}{2}, 1 \right) \cup \left( 1, \frac{3}{2} \right) \]