Let's break down the expression from the image you uploaded:

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

Step 1: Apply the logarithmic property

Using the property of logarithms that states:

$\log_b a + \log_b c = \log_b (a \cdot c)$

We can combine the two logarithms:

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

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, cancel out the $y$ terms:

$\frac{16x^3y}{y} = 16x^3$

Step 3: Final result

Thus, the expression simplifies to:

$\log_2 (16x^3)$

Step 4: Simplify further

We can express $16$ as $2^4$, so:

$\log_2 (2^4 x^3)$

Using the logarithmic property $\log_b (ab) = \log_b a + \log_b b$, we can separate this:

$\log_2 (2^4) + \log_2 (x^3)$

Now simplify:

$\log_2 (2^4) = 4$ $\log_2 (x^3) = 3\log_2 x$

So, the expression becomes:

$4 + 3\log_2 x$

Final Answer:

The simplified expression is:

$4 + 3\log_2 x$

Would you like further clarification or additional steps? Here are some follow-up questions to explore:

1. How can we simplify logarithmic expressions involving multiple terms and bases?
2. What are the logarithmic properties used in solving this type of problem?
3. Can you solve logarithmic equations where the argument is a product or quotient?
4. How do you apply the change of base formula to logarithmic expressions?
5. What is the significance of the logarithmic identity $\log_b a^n = n \log_b a$?

Tip: Always remember to apply the logarithmic properties carefully, such as combining logs, using the power rule, and simplifying the arguments within the logarithms for easier calculations.