Let ` f(x) = log _({x}) [x]`
` g (x) =log _({x})-{x}`
`h (x) log _({x}) {x}`
where `[], {}` denotes the greatest integer function and fractional part function respectively.
For `x in (1,5)the f (x)` is not defined at how many points :

Options

A. 5
B. 4
C. 3
D. 2

Answer

To solve the problem, we need to determine the points in the interval (1, 5) where the function \( f(x) = \log_{ \{x\} } [x] \) is not defined.

### Step-by-step Solution:

1. **Understanding the Function**:
The function \( f(x) \) is defined as \( f(x) = \log_{ \{x\} } [x] \), where \( [x] \) is the greatest integer function (floor function) and \( \{x\} \) is the fractional part function.

2. **Identifying the Interval**:
We are interested in the interval \( (1, 5) \). This means we will consider values of \( x \) that are greater than 1 and less than 5.

3. **Conditions for the Logarithm**:
The logarithm \( \log_{ \{x\} } [x] \) is defined under two conditions:
- The base \( \{x\} \) must be greater than 0 and not equal to 1.
- The argument \( [x] \) must be greater than 0.

4. **Analyzing the Base \( \{x\} \)**:
The fractional part \( \{x\} = x - [x] \) is always between 0 and 1 (0 ≤ \( \{x\} \) < 1). The base \( \{x\} \) will be equal to 0 when \( x \) is an integer. Thus, \( f(x) \) will not be defined at integer values of \( x \).

5. **Identifying Integer Points in the Interval**:
The integers in the interval \( (1, 5) \) are 2, 3, and 4. At these points, \( \{x\} = 0 \), which makes the logarithm undefined.

6. **Conclusion**:
Therefore, the function \( f(x) \) is not defined at 3 points: \( x = 2, 3, \) and \( 4 \).

### Final Answer:
The function \( f(x) \) is not defined at **3 points** in the interval \( (1, 5) \).

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