The expression provided is:

$2 \log_4 x + 2 \left( \log_4 w - 2 \log_4 y \right)$

Let’s simplify this step by step:

Step 1: Distribute the 2 in the second term

We can distribute the $2$ across the parentheses:

$2 \log_4 x + 2 \log_4 w - 4 \log_4 y$

Step 2: Use logarithmic properties

We apply the property $a \log_b c = \log_b c^a$ to the coefficients:

$\log_4 x^2 + \log_4 w^2 - \log_4 y^4$

Step 3: Combine the logarithms using addition and subtraction rules

Using the logarithmic rules $\log_b A + \log_b B = \log_b (A \cdot B)$ and $\log_b A - \log_b B = \log_b \left( \frac{A}{B} \right)$, we can simplify the expression:

$\log_4 \left( \frac{x^2 w^2}{y^4} \right)$

Final Simplified Expression:

The expression simplifies to:

$\log_4 \left( \frac{x^2 w^2}{y^4} \right)$

Would you like further clarification or any more details?

Here are five related questions to deepen your understanding:

1. How can you apply the power rule of logarithms in other contexts?
2. What is the general rule for simplifying logarithmic expressions with different bases?
3. Can this simplified logarithmic expression be used to solve an equation involving $x$, $w$, and $y$?
4. How would this expression change if the logarithmic base were changed from 4 to 10?
5. What happens to the expression if one of the variables (e.g., $y$) equals 1?

Tip: Always double-check if you can simplify logarithmic terms further using properties like the product, quotient, and power rules.