Determine the smallest integer that is in the domain of log2(log3(log4(log5(x)))). 

We are tasked with finding the smallest integer \( x \) that is in the domain of the function:

\[
f(x) = \log_2(\log_3(\log_4(\log_5(x))))
\]

### Step 1: Understanding the Domain
The logarithmic function \( \log_b(y) \) is defined only for \( y > 0 \). Thus, we need to ensure that the arguments of all the nested logarithms are positive.

### Step 2: Working from the Inside Out
We will begin with the innermost logarithm and work our way outward:

1. **Innermost logarithm**: \( \log_5(x) \)
   - For \( \log_5(x) \) to be defined, we need \( x > 5 \).

2. **Next logarithm**: \( \log_4(\log_5(x)) \)
   - For \( \log_4(y) \) to be defined, \( y > 0 \). In this case, \( y = \log_5(x) \), so we require:
     \[
     \log_5(x) > 0
     \]
     This inequality is true when \( x > 5^0 = 1 \). However, since \( x > 5 \) from the previous step, this condition is already satisfied.

3. **Third logarithm**: \( \log_3(\log_4(\log_5(x))) \)
   - Again, for \( \log_3(y) \) to be defined, \( y > 0 \). Here, \( y = \log_4(\log_5(x)) \), so we require:
     \[
     \log_4(\log_5(x)) > 0
     \]
     This inequality holds when:
     \[
     \log_5(x) > 4^0 = 1
     \]
     Thus, we need \( \log_5(x) > 1 \), which implies \( x > 5^1 = 5 \). Again, this is consistent with the condition \( x > 5 \) from earlier.

4. **Outermost logarithm**: \( \log_2(\log_3(\log_4(\log_5(x)))) \)
   - Finally, for \( \log_2(y) \) to be defined, \( y > 0 \). In this case, \( y = \log_3(\log_4(\log_5(x))) \), so we require:
     \[
     \log_3(\log_4(\log_5(x))) > 0
     \]
     This inequality holds when:
     \[
     \log_4(\log_5(x)) > 3^0 = 1
     \]
     From this, we need \( \log_5(x) > 4^1 = 4 \), which implies \( x > 5^4 = 625 \).

### Step 3: Conclusion
The smallest integer \( x \) that satisfies all the conditions is \( x = 625 \). Therefore, the smallest integer in the domain of the function is:

\[
\boxed{625}
\]

Would you like a more detailed breakdown or have any specific questions about the process?

Here are 5 related questions to expand your understanding:
1. What is the domain of \( \log_b(x) \) for any base \( b \)?
2. How does the base of a logarithm affect the solution of inequalities?
3. Can you solve a similar problem with a different set of nested logarithms, such as \( \log_3(\log_2(x)) \)?
4. What happens if you replace one of the logarithmic bases with a fractional base, like \( \log_{1/2}(x) \)?
5. How would the answer change if we changed the base of the outermost logarithm?

**Tip**: When solving domain problems involving logarithms, always remember that the argument of a logarithmic function must be greater than zero!

Determine the smallest integer that is in the domain of log2(log3(log4(log5(x)))).

Solve the problem of finding the smallest integer in the domain of the function log2(log3(log4(log5(x)))). This problem explores nested logarithms, domain constraints, and logarithmic inequalities, suitable for high school students in Grades 10-12.

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