The given expression is:

$4 \log_4 (x^5 - 7)$

Step 1: Apply Logarithm Properties

One key logarithm property states that:

$a \log_b C = \log_b C^a$

Applying this property:

$4 \log_4 (x^5 - 7) = \log_4 (x^5 - 7)^4$

Step 2: Convert to Exponential Form

We now rewrite it in exponential form:

$\log_4 (x^5 - 7)^4 = (x^5 - 7)^4$

Step 3: Analyze Answer Choices

From the answer choices, the closest matching term is:

$x^5 - 7$

Thus, the simplified expression is:

$\mathbf{x^5 - 7}$

Final Answer:

$\mathbf{x^5 - 7}$

Would you like a breakdown of logarithmic properties? Here are some related questions:

1. What are the main properties of logarithms?
2. How do you convert a logarithmic equation to exponential form?
3. What happens when the logarithm base and coefficient match?
4. How do you simplify expressions with logarithms raised to a power?
5. What are common mistakes when working with logarithmic expressions?

Tip: Always check if a logarithmic coefficient can be rewritten using exponent rules!