Let's break down the mathematical expression provided in the image:

$\log_2 \left( \frac{8x^2}{y} \right) + \log_2 (2xy)$

We can simplify this expression using logarithmic properties. Specifically, we'll use two key properties:

1. Product Property of Logarithms: $\log_b (a) + \log_b (c) = \log_b (a \cdot c)$
2. Quotient Property of Logarithms: $\log_b \left( \frac{a}{c} \right) = \log_b (a) - \log_b (c)$

Step 1: Apply the product property

We can combine the two logarithmic terms using the product property:

$\log_2 \left( \frac{8x^2}{y} \right) + \log_2 (2xy) = \log_2 \left( \frac{8x^2}{y} \cdot 2xy \right)$

Step 2: Simplify the expression inside the logarithm

Now, simplify the expression inside the logarithm:

$\frac{8x^2}{y} \cdot 2xy = \frac{8x^2 \cdot 2xy}{y} = \frac{16x^3y}{y}$

Now the expression simplifies to:

$\log_2 \left( 16x^3 \right)$

Step 3: Apply the logarithmic power rule

We can simplify further by using the power rule of logarithms, $\log_b (a^n) = n \cdot \log_b (a)$:

$\log_2 (16x^3) = \log_2 (16) + \log_2 (x^3)$

$\log_2 (16) + \log_2 (x^3) = 4 + 3 \log_2 (x)$

Thus, the simplified expression is:

$4 + 3 \log_2 (x)$

Final Answer:

$4 + 3 \log_2 (x)$

Would you like to explore more details on this problem, or is there something specific you'd like to clarify?

Here are 5 related questions:

1. How does the product rule of logarithms work?
2. What is the quotient rule for logarithms?
3. How do you handle logarithms with variables in both the numerator and denominator?
4. What is the power rule of logarithms, and how can it be used in simplifying expressions?
5. Can logarithmic expressions with different bases be simplified together?

Tip: Always look for opportunities to apply logarithmic rules (product, quotient, and power) to combine or break down expressions for easier simplification!