`f(x)=log(x-[x])`, where `[*]` denotes the greatest integer function. find the domain of f(x).

Answer

To find the domain of the function \( f(x) = \log(x - [x]) \), where \([x]\) denotes the greatest integer function, we need to ensure that the argument of the logarithm is positive.

### Step-by-Step Solution:

1. **Understanding the Greatest Integer Function**:
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). We can express \(x\) as:
\[ x = n + f \]
where \(n = [x]\) (an integer) and \(f\) is the fractional part of \(x\) such that \(0 \leq f < 1\).

2. **Expressing the Function**:
We can rewrite the expression inside the logarithm:
\[ x - [x] = (n + f) - n = f \]
Therefore, we have:
\[ f(x) = \log(f) \]

3. **Finding the Condition for the Logarithm**:
For the logarithm to be defined, the argument must be positive:
\[ f > 0 \]
This means:
\[ x - [x] > 0 \]

4. **Analyzing the Fractional Part**:
The fractional part \(f\) is defined as:
\[ f = x - [x] \]
The condition \(f > 0\) implies that:
\[ x - [x] > 0 \implies x \text{ is not an integer} \]

5. **Determining the Domain**:
Since \(x\) cannot be an integer, the domain of \(f(x)\) is all real numbers except integers. Therefore, we can express the domain as:
\[ \text{Domain of } f(x) = \mathbb{R} \setminus \mathbb{Z} \]

### Final Answer:
The domain of \( f(x) = \log(x - [x]) \) is:
\[ \mathbb{R} \setminus \mathbb{Z} \]