The domain of the function f(x)=`1/sqrt(log_10x` is :

Options

A. R
B. `R^+-(0,1]`
C. R-{0}
D. none of these

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{\log_{10} x}} \), we need to determine the values of \( x \) for which this function is defined and real.

### Step-by-Step Solution:

1. **Identify the logarithmic function**: The function involves \( \log_{10} x \). The logarithm is only defined for positive values of \( x \). Therefore, we have:
\[ x > 0 \]

2. **Set the condition for the square root**: The expression inside the square root, \( \log_{10} x \), must be greater than zero for the square root to be defined and real. Thus, we need:
\[ \log_{10} x > 0 \]

3. **Solve the logarithmic inequality**: The inequality \( \log_{10} x > 0 \) means that \( x \) must be greater than \( 10^0 \):
\[ x > 1 \]

4. **Combine the conditions**: From the above steps, we have two conditions:
- \( x > 0 \) (from the definition of logarithm)
- \( x > 1 \) (from the square root condition)

Since \( x > 1 \) is a stronger condition than \( x > 0 \), we can conclude that the domain of the function is:
\[ x > 1 \]

5. **Express the domain in interval notation**: The domain can be expressed in interval notation as:
\[ (1, \infty) \]

### Conclusion:
The domain of the function \( f(x) = \frac{1}{\sqrt{\log_{10} x}} \) is \( (1, \infty) \).

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