The domain of f(x)=`log_(10)` (sin x) contains which of the following intervals?

Options

A. `0lexlepi`
B. `-pi/2lexlepi/2`
C. `0ltxltpi`
D. `-pi/2ltxltpi/2`

Explanation

To find the domain of the function \( f(x) = \log_{10}(\sin x) \), we need to determine the values of \( x \) for which \( f(x) \) is defined. The logarithmic function is defined only for positive arguments, so we need to ensure that \( \sin x > 0 \).

### Step 1: Determine when \( \sin x > 0 \)
The sine function is positive in the first and second quadrants. This means that:
- In the interval \( (0, \pi) \), \( \sin x \) is positive.
- The sine function is zero at \( x = n\pi \) for any integer \( n \). Therefore, we must exclude these points from our domain.

### Step 2: Identify the intervals
Since \( \sin x \) is positive in the interval \( (0, \pi) \), we can express the domain of \( f(x) \) as:
- \( x \in (0, \pi) \)

### Step 3: Exclude points where \( \sin x = 0 \)
We need to ensure that we exclude the endpoints where \( \sin x = 0 \):
- At \( x = 0 \) and \( x = \pi \), \( \sin x = 0 \), which is not allowed since \( \log(0) \) is undefined.

### Conclusion
Thus, the domain of the function \( f(x) = \log_{10}(\sin x) \) is:
- \( (0, \pi) \)

### Final Answer
The domain of \( f(x) = \log_{10}(\sin x) \) contains the interval \( (0, \pi) \).

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