The domain of the function f(x)=`logx^2` is :

Options

A. R
B. `R^+`
C. R-{0}
D. `R^+cup{0}`

Answer

To find the domain of the function \( f(x) = \log(x^2) \), we need to determine the values of \( x \) for which the function is defined. The logarithmic function is defined under certain conditions.

### Step-by-step Solution:

1. **Identify the logarithmic function**:
The function given is \( f(x) = \log(x^2) \).

2. **Understand the conditions for the logarithm**:
The logarithm \( \log_b(a) \) is defined when:
- \( a > 0 \) (the argument of the logarithm must be positive)
- \( b > 0 \) (the base of the logarithm must be positive)
- \( b \neq 1 \) (the base cannot be 1)

In this case, we are considering \( \log(x^2) \) where the base is assumed to be 10 (or any valid base greater than 0 and not equal to 1).

3. **Set the argument greater than zero**:
For the function \( \log(x^2) \) to be defined, we need:
\[ x^2 > 0 \]

4. **Solve the inequality**:
The expression \( x^2 > 0 \) is true for all real numbers \( x \) except when \( x = 0 \). Thus, the only value that makes \( x^2 \) equal to zero is \( x = 0 \).

5. **Determine the domain**:
Therefore, the domain of the function \( f(x) = \log(x^2) \) is all real numbers except \( x = 0 \). In interval notation, this can be expressed as:
\[ (-\infty, 0) \cup (0, \infty) \]

### Final Answer:
The domain of the function \( f(x) = \log(x^2) \) is \( \mathbb{R} \setminus \{0\} \) or in interval notation, \( (-\infty, 0) \cup (0, \infty) \).

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