`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 analyze the expression inside the logarithm.

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression \( x - [x] \) represents the fractional part of \( x \). This fractional part is defined as:
\[ x - [x] = \{x\} \]
where \(\{x\}\) is the fractional part of \( x \).

2. **Properties of the Fractional Part**:
The fractional part \(\{x\}\) has the following properties:
- It is always in the range \( [0, 1) \).
- It equals \( 0 \) when \( x \) is an integer.

3. **Logarithm Domain Requirement**:
The logarithm function, \(\log(y)\), is defined only for \( y > 0 \). Therefore, we need:
\[ x - [x] > 0 \]
This condition implies that:
\[ \{x\} > 0 \]

4. **Finding Values of \( x \)**:
From the properties of the fractional part, we know that \(\{x\} = 0\) when \( x \) is an integer. Thus, the condition \(\{x\} > 0\) holds true for all \( x \) that are not integers.

5. **Conclusion**:
Therefore, the domain of the function \( f(x) \) is all real numbers except the integers. In interval notation, this can be expressed as:
\[ \text{Domain of } f(x) = \mathbb{R} \setminus \mathbb{Z} \]

### Final Answer:
The domain of \( f(x) = \log(x - [x]) \) is all real numbers except integers, or \( \mathbb{R} \setminus \mathbb{Z} \).