To expand the given logarithmic expression $\log \left( \frac{y^4 x}{z^4} \right)$ fully using the properties of logarithms, let's apply the following logarithmic rules:

1. Logarithm of a quotient:
   $\log \left( \frac{A}{B} \right) = \log A - \log B$

2. Logarithm of a product:
   $\log (A B) = \log A + \log B$

3. Logarithm of a power:
   $\log (A^n) = n \log A$

Now, let's apply these to the expression:

Step 1: Apply the quotient rule.

We have $\log \left( \frac{y^4 x}{z^4} \right)$, so:

$\log \left( \frac{y^4 x}{z^4} \right) = \log(y^4 x) - \log(z^4)$

Step 2: Apply the product rule to $\log(y^4 x)$.

$\log(y^4 x) = \log(y^4) + \log(x)$

Step 3: Apply the power rule to $\log(y^4)$ and $\log(z^4)$.

$\log(y^4) = 4 \log(y), \quad \log(z^4) = 4 \log(z)$

Step 4: Combine all terms.

Now, we can substitute all the terms back into the original expression:

$\log \left( \frac{y^4 x}{z^4} \right) = 4 \log(y) + \log(x) - 4 \log(z)$

Final Answer:

$\boxed{4 \log(y) + \log(x) - 4 \log(z)}$

This is the fully expanded form of the logarithmic expression.

Would you like a more detailed explanation or have any questions on this? Here are some follow-up questions for you:

1. How do we apply the logarithmic properties to an expression like $\log \left( \frac{x^2 y}{z^3} \right)$?
2. What happens if we have a logarithmic expression with multiple terms like $\log(xy) + \log(z)$?
3. Can you expand $\log \left( \frac{x^3 y^2}{z} \right)$ using the same rules?
4. How do we simplify a logarithmic expression when the base is different from 10 or $e$?
5. What are the key properties of logarithms when dealing with exponential functions?

Tip: Always use the properties of logarithms step-by-step to break down complex expressions, and remember to simplify as you go!