Which one of the following is true for `x in R:`

Options

A. `f(x)=log|x|`
B. `f(x)=(|x|)/(log|x|)`
C. `f(x)=log(x^(2)+2)`
D. `f(x)=log_(x)100`

Answer

To determine which of the given options is true for \( x \in \mathbb{R} \), we will analyze each option step by step.

### Step 1: Analyze Option 1 - \( \log |x| \)
- The function \( \log |x| \) is defined for \( |x| > 0 \), which means \( x \) cannot be 0.
- Therefore, the domain of \( \log |x| \) is \( (-\infty, 0) \cup (0, \infty) \).
- Since \( x = 0 \) is not included in the domain, this option is **not true**.

### Step 2: Analyze Option 2 - \( \log |x| \) (again)
- This option is essentially the same as Option 1.
- As established, \( \log |x| \) is undefined at \( x = 0 \).
- Thus, this option is also **not true**.

### Step 3: Analyze Option 3 - \( \log(x^2 + 2) \)
- The expression inside the logarithm is \( x^2 + 2 \).
- Since \( x^2 \) is always non-negative, \( x^2 + 2 \) is always greater than 0 for any real number \( x \).
- Therefore, \( \log(x^2 + 2) \) is defined for all \( x \in \mathbb{R} \).
- This option is **true**.

### Step 4: Analyze Option 4 - \( \log_{x}(100) \)
- The function \( \log_{x}(100) \) is defined for \( x > 0 \) and \( x \neq 1 \).
- This means \( x \) cannot be 0 or 1.
- Since we can have \( x = 1 \) or \( x = 0 \) in the real numbers, this option is **not true**.

### Conclusion
After analyzing all options, we find that **Option 3** \( \log(x^2 + 2) \) is the only option that is true for \( x \in \mathbb{R} \).

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