The domain of definitions of `f(x)= underset(to n" times "leftrightarrow)(log_(10) log_(10)......log_(10)x)" is: :`

Options

A. `(10^(n),oo)`
B. `(10^(n-1),oo)`
C. `(10^(n-2),oo)`
D. None of these

Answer

To find the domain of the function \( f(x) = \log_{10}(\log_{10}(\ldots(\log_{10} x)\ldots)) \) where the logarithm is applied \( n \) times, we need to ensure that each logarithmic function is defined and positive.

### Step-by-Step Solution:

1. **Understanding the Logarithm**:
The logarithm \( \log_{10}(x) \) is defined only for \( x > 0 \) and is positive when \( x > 1 \).

2. **First Iteration (n=1)**:
- For \( n = 1 \), we have \( f(x) = \log_{10}(x) \).
- The condition for \( f(x) \) to be defined is \( x > 0 \).
- The condition for \( f(x) \) to be positive is \( x > 1 \).
- Therefore, for \( n = 1 \), the domain is \( x > 1 \).

3. **Second Iteration (n=2)**:
- For \( n = 2 \), we have \( f(x) = \log_{10}(\log_{10}(x)) \).
- The inner logarithm \( \log_{10}(x) \) must be positive, which means \( \log_{10}(x) > 0 \) or \( x > 10^0 = 1 \).
- Thus, \( x > 1 \) is a necessary condition.
- Additionally, \( \log_{10}(x) \) must also be greater than 1 for the outer logarithm to be defined, which gives us \( x > 10^1 = 10 \).
- Therefore, for \( n = 2 \), the domain is \( x > 10 \).

4. **Third Iteration (n=3)**:
- For \( n = 3 \), we have \( f(x) = \log_{10}(\log_{10}(\log_{10}(x))) \).
- The innermost logarithm \( \log_{10}(x) \) must be greater than 1, giving \( x > 10 \).
- The next logarithm \( \log_{10}(\log_{10}(x)) \) must also be greater than 0, which means \( \log_{10}(x) > 10^0 = 1 \) or \( x > 10 \).
- Finally, \( \log_{10}(\log_{10}(x)) \) must also be greater than 1, which gives \( \log_{10}(x) > 10^1 = 10 \), leading to \( x > 10^{10} \).
- Therefore, for \( n = 3 \), the domain is \( x > 10^{10} \).

5. **General Pattern**:
- Continuing this pattern, for \( n \) iterations, we find that the domain becomes \( x > 10^{10^{(n-1)}} \).
- This means that for each additional logarithm, the lower bound increases exponentially.

### Conclusion:
The domain of the function \( f(x) = \log_{10}(\log_{10}(\ldots(\log_{10} x)\ldots)) \) applied \( n \) times is:
\[ \text{Domain: } x > 10^{10^{(n-1)}} \]