The domain of definition of the function `f(X)=x^(log_(10)x`, is

Options

A. `(0,1) cup (1,oo)`
B. `(0,oo)`
C. `[1,oo)`
D. `[0,1) cup (1,oo)`

Answer

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

### Step-by-Step Solution:

1. **Understanding the Function**: The function is defined as \( f(x) = x^{\log_{10} x} \). This involves both the base \( x \) and the exponent \( \log_{10} x \).

2. **Conditions for the Base**: The base \( x \) must be positive because the logarithm function \( \log_{10} x \) is only defined for \( x > 0 \). Therefore, we have:
\[ x > 0 \]

3. **Conditions for the Exponent**: The exponent \( \log_{10} x \) must also be defined. Since \( \log_{10} x \) is defined for \( x > 0 \), we already satisfy this condition with our previous step.

4. **Exclusion of Specific Values**: We also need to consider if there are any specific values of \( x \) that could cause issues with the function. The expression \( x^{\log_{10} x} \) is well-defined as long as \( x \neq 1 \) because:
- If \( x = 1 \), then \( \log_{10} 1 = 0 \) and \( 1^0 = 1 \), which is defined but does not contribute to the exponential nature of the function.
- However, since we are looking for a domain where the function behaves like an exponential function, we exclude \( x = 1 \).

5. **Conclusion**: Therefore, the domain of the function \( f(x) = x^{\log_{10} x} \) is:
\[ x > 0 \quad \text{and} \quad x \neq 1 \]
This can be expressed in interval notation as:
\[ (0, 1) \cup (1, \infty) \]

### Final Answer:
The domain of the function \( f(x) = x^{\log_{10} x} \) is \( (0, 1) \cup (1, \infty) \).