6) $$ \log _{4}\left(z^{4} \sqrt{x}\right) $$

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To simplify the expression $$ \log_{4}\left(z^{4} \sqrt{x}\right) $$, we can use the properties of logarithms:

1. **Product Rule**: $$ \log_b (A \cdot B) = \log_b A + \log_b B $$
2. **Power Rule**: $$ \log_b (A^k) = k \log_b A $$
3. **Square Root as Exponent**: $$ \sqrt{x} = x^{1/2} $$

Let's apply these step by step:

1. **Express the Square Root as an Exponent**:
   $$
   \sqrt{x} = x^{1/2}
   $$
   So, the original expression becomes:
   $$
   \log_{4}\left(z^{4} \cdot x^{1/2}\right)
   $$

2. **Apply the Product Rule**:
   $$
   \log_{4}\left(z^{4}\right) + \log_{4}\left(x^{1/2}\right)
   $$

3. **Apply the Power Rule to Each Term**:
   $$
   4 \log_{4} z + \frac{1}{2} \log_{4} x
   $$

So, the simplified form of the expression is:

$$
\log_{4}\left(z^{4} \sqrt{x}\right) = 4 \log_{4} z + \frac{1}{2} \log_{4} x
$$

**Final Answer:**
After expanding, the expression is equal to 4 log₄ z plus one half log₄ x. Thus,

4 log₄ z + ½ log₄ x
The simplified form of the expression is $$ 4 \log_{4} z + \frac{1}{2} \log_{4} x $$.

6) \( \log _{4}\left(z^{4} \sqrt{x}\right) \)

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