The expression in the image is:

$\log_2 \left( x^5 (x + 2) \right)$

To simplify or evaluate this logarithmic expression, we can apply the following logarithmic properties:

1. Product Rule: $\log_b(mn) = \log_b(m) + \log_b(n)$
2. Power Rule: $\log_b(m^k) = k \cdot \log_b(m)$

Step 1: Apply the product rule

$\log_2 \left( x^5 (x + 2) \right) = \log_2(x^5) + \log_2(x + 2)$

Step 2: Apply the power rule to $\log_2(x^5)$

$\log_2(x^5) = 5 \cdot \log_2(x)$

Thus, the expression becomes: $5 \cdot \log_2(x) + \log_2(x + 2)$

Final Simplified Form:

$\boxed{5 \cdot \log_2(x) + \log_2(x + 2)}$

Let me know if you want further clarification or details!

Here are 5 related questions for exploration:

1. How do you solve for $x$ if the entire logarithmic expression equals a number?
2. What is the domain of the given logarithmic function?
3. How do logarithmic rules extend to base changes?
4. How would this simplify if $x = 2$?
5. What happens if we replace the base $2$ with a different base?

Tip: Always verify the domain of a logarithmic expression to ensure that the arguments are positive and valid!