To solve this logarithmic expression $\log_b (y^5x)$, we will apply the properties of logarithms.

Steps:

1. Use the product rule of logarithms, which states: $\log_b (AB) = \log_b A + \log_b B$ Applying this to $\log_b (y^5x)$: $\log_b (y^5x) = \log_b (y^5) + \log_b x$

2. Use the power rule of logarithms, which states: $\log_b (A^n) = n \log_b A$ Applying this to $\log_b (y^5)$: $\log_b (y^5) = 5 \log_b y$

3. Putting everything together: $\log_b (y^5x) = 5 \log_b y + \log_b x$

So, the expanded form is: $\log_b (y^5x) = 5 \log_b y + \log_b x$

Let me know if you'd like further details or have any questions!

Related Questions:

1. How do the product and quotient rules for logarithms differ?
2. How can the change of base formula be applied to logarithms?
3. What is the inverse function of logarithms?
4. Can you simplify $\log_b (x^2 / y^3)$?
5. How do logarithmic properties apply to solving equations?

Tip:

Always remember the key logarithmic rules (product, quotient, and power) as they are useful in simplifying complex expressions!