To find the domain of the function \( f(x) = \log_{10}(\log_{10}(\log_{10}(\log_{10}(\log_{10} x)))) \), we need to ensure that each logarithmic function is defined and positive.
### Step-by-Step Solution:
1. **Innermost Logarithm**: Start with the innermost logarithm, which is \( \log_{10} x \).
- For \( \log_{10} x \) to be defined, \( x \) must be greater than 0.
- For \( \log_{10} x \) to be positive, we need \( \log_{10} x > 0 \).
- This implies \( x > 10^0 = 1 \).
2. **Second Logarithm**: Now consider \( \log_{10}(\log_{10} x) \).
- For this to be defined, \( \log_{10} x \) must be greater than 0, which we already established means \( x > 1 \).
- For \( \log_{10}(\log_{10} x) \) to be positive, we need \( \log_{10} x > 10^0 = 1 \).
- This implies \( x > 10^1 = 10 \).
3. **Third Logarithm**: Next, consider \( \log_{10}(\log_{10}(\log_{10} x)) \).
- For this to be defined, \( \log_{10}(\log_{10} x) \) must be greater than 0, which means \( \log_{10} x > 10^1 = 10 \).
- This implies \( x > 10^{10} \).
4. **Fourth Logarithm**: Now consider \( \log_{10}(\log_{10}(\log_{10}(\log_{10} x))) \).
- For this to be defined, \( \log_{10}(\log_{10}(\log_{10} x)) \) must be greater than 0, which means \( \log_{10}(\log_{10} x) > 10^1 = 10 \).
- This implies \( \log_{10} x > 10^{10} \), leading to \( x > 10^{10^{10}} \).
5. **Outermost Logarithm**: Finally, consider \( \log_{10}(\log_{10}(\log_{10}(\log_{10}(\log_{10} x)))) \).
- For this to be defined, \( \log_{10}(\log_{10}(\log_{10}(\log_{10} x))) \) must be greater than 0, which means \( \log_{10}(\log_{10}(\log_{10} x)) > 10^1 = 10 \).
- This implies \( \log_{10}(\log_{10} x) > 10^{10} \), leading to \( \log_{10} x > 10^{10^{10}} \), which means \( x > 10^{10^{10^{10}}} \).
### Conclusion:
The domain of the function \( f(x) = \log_{10}(\log_{10}(\log_{10}(\log_{10}(\log_{10} x)))) \) is:
\[
\text{Domain} = (10^{10^{10^{10}}}, \infty)
\]
