Let's expand the given logarithmic expression using the properties of logarithms.

The expression is:

$\log_2 \left( \frac{m^5 n^3 n^8}{p^4} \right)$

We will apply the following properties of logarithms:

1. Logarithm of a quotient:
   $\log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y$

2. Logarithm of a product:
   $\log_b (xy) = \log_b x + \log_b y$

3. Power rule:
   $\log_b (x^n) = n \log_b x$

Step 1: Combine $n^3$ and $n^8$

We can simplify the expression inside the logarithm:

$\frac{m^5 n^3 n^8}{p^4} = \frac{m^5 n^{3+8}}{p^4} = \frac{m^5 n^{11}}{p^4}$

Step 2: Apply the logarithm of a quotient rule

$\log_2 \left( \frac{m^5 n^{11}}{p^4} \right) = \log_2 (m^5 n^{11}) - \log_2 (p^4)$

Step 3: Apply the logarithm of a product rule to the first term

$\log_2 (m^5 n^{11}) = \log_2 (m^5) + \log_2 (n^{11})$

Step 4: Apply the power rule to each term

$\log_2 (m^5) = 5 \log_2 m \quad \text{and} \quad \log_2 (n^{11}) = 11 \log_2 n \quad \text{and} \quad \log_2 (p^4) = 4 \log_2 p$

Final Expression:

$5 \log_2 m + 11 \log_2 n - 4 \log_2 p$

Thus, the correct expanded form of the logarithmic expression is:

$5 \log_2 m + 11 \log_2 n - 4 \log_2 p$

From the options in the image, this corresponds to the second choice:

$5 \log_2 m + 11 \log_2 n - 4 \log_2 p$

Would you like further details or have any questions?

Here are some related questions you might find helpful:

1. How can we apply logarithmic rules to simplify different types of expressions?
2. What are the other common properties of logarithms?
3. Can logarithms be applied to complex numbers, and how?
4. How do logarithmic identities relate to exponential functions?
5. What is the change of base formula for logarithms?

Tip: When expanding logarithmic expressions, always look for products, quotients, and powers to apply the logarithmic properties efficiently.