The problem asks to rewrite the expression $\log_b (8xy)$ in terms of the logarithms of $x$ and $y$. Using the properties of logarithms, we can break this down step by step:

Solution:

1. Use the product rule of logarithms:
   $\log_b (8xy) = \log_b (8) + \log_b (x) + \log_b (y)$

2. Simplify further:
   If the logarithm of 8 can be expressed in another form, such as in terms of powers, we could rewrite it, but for now, it remains $\log_b (8)$.

Final Answer:

$\log_b (8xy) = \log_b (8) + \log_b (x) + \log_b (y)$

Let me know if you'd like to explore this further or solve similar problems!

Here are five follow-up questions:

1. What is the logarithm of a number raised to a power?
2. Can you express $\log_b(8)$ in terms of simpler logarithms (e.g., base 2)?
3. How do logarithmic properties apply to division (e.g., $\log_b \frac{x}{y}$)?
4. What are some real-world applications of these logarithmic properties?
5. Can you rewrite $\log_b (x^2y^3)$ using the properties of logarithms?

Tip:

Always remember to use the product, quotient, and power rules of logarithms to break down complex logarithmic expressions!