We are tasked with finding \( f'(x) \), where \( f(x) = \left( \log(x^5) \right)^4 \).

### Steps:

1. **Simplify the expression**: 
   First, use the logarithmic identity \( \log(a^b) = b \log(a) \) to simplify:
   \[
   f(x) = \left( 5 \log(x) \right)^4
   \]

2. **Apply the chain rule**:
   Let \( u(x) = 5 \log(x) \), so \( f(x) = u(x)^4 \).
   
   Using the chain rule, \( f'(x) = 4 u(x)^3 \cdot u'(x) \).

3. **Differentiate \( u(x) = 5 \log(x) \)**:
   The derivative of \( u(x) \) with respect to \( x \) is:
   \[
   u'(x) = 5 \cdot \frac{1}{x} = \frac{5}{x}
   \]

4. **Substitute \( u(x) \) and \( u'(x) \) into the derivative**:
   \[
   f'(x) = 4 \left( 5 \log(x) \right)^3 \cdot \frac{5}{x}
   \]

5. **Simplify the final expression**:
   \[
   f'(x) = \frac{4 \cdot 5^4 \cdot (\log(x))^3}{x} = \frac{2500 \cdot (\log(x))^3}{x}
   \]

Thus, the derivative is:
\[
f'(x) = \frac{2500 (\log(x))^3}{x}
\]

Would you like further details or have any questions?

### Relative questions:
1. How does the chain rule apply to composite functions?
2. What is the logarithmic differentiation technique?
3. Can you explain the relationship between exponential and logarithmic functions in calculus?
4. How would you approach the derivative of a product of two logarithmic functions?
5. What are the typical applications of logarithmic differentiation in solving real-world problems?

### Tip:
Always simplify the expression as much as possible before applying differentiation rules. It often makes the process smoother and reduces potential mistakes.

Explore the step-by-step solution to find the derivative of f(x) = (log(x^5))^4. This problem covers logarithmic properties and the application of the chain rule in differentiation, ideal for calculus students. The final solution simplifies to f'(x) = (2500 * (log(x))^3) / x.

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