Use the properties of logarithms to expand the following expression. $$ \log \sqrt[3]{\frac{x y^{4}}{z^{2}}} $$ Each logarithm should involve only one variable and should not have any radicals or exponent You may assume that all variables are positive. $$ \log _{\sqrt[3]{ }}^{\frac{x y^{4}}{z^{2}}}=\square $$

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To expand the expression $$ \log \sqrt[3]{\frac{x y^{4}}{z^{2}}} $$ using the properties of logarithms, we can follow these steps:

1. **Rewrite the expression**: The cube root can be expressed as an exponent. Thus, we have:
   $$
   \log \sqrt[3]{\frac{x y^{4}}{z^{2}}} = \log \left( \frac{x y^{4}}{z^{2}} \right)^{\frac{1}{3}}
   $$

2. **Apply the power rule of logarithms**: The power rule states that $$ \log(a^b) = b \cdot \log(a) $$. Applying this rule gives us:
   $$
   \log \left( \frac{x y^{4}}{z^{2}} \right)^{\frac{1}{3}} = \frac{1}{3} \log \left( \frac{x y^{4}}{z^{2}} \right)
   $$

3. **Use the quotient rule of logarithms**: The quotient rule states that $$ \log\left(\frac{a}{b}\right) = \log(a) - \log(b) $$. We can apply this to our expression:
   $$
   \frac{1}{3} \log \left( \frac{x y^{4}}{z^{2}} \right) = \frac{1}{3} \left( \log(x y^{4}) - \log(z^{2}) \right)
   $$

4. **Apply the product rule of logarithms**: The product rule states that $$ \log(ab) = \log(a) + \log(b) $$. We can apply this to $$ \log(x y^{4}) $$:
   $$
   \log(x y^{4}) = \log(x) + \log(y^{4})
   $$

5. **Apply the power rule again**: For $$ \log(y^{4}) $$, we can use the power rule:
   $$
   \log(y^{4}) = 4 \log(y)
   $$

6. **Combine everything**: Now we can substitute back into our expression:
   $$
   \frac{1}{3} \left( \log(x) + 4 \log(y) - \log(z^{2}) \right)
   $$
   And for $$ \log(z^{2}) $$:
   $$
   \log(z^{2}) = 2 \log(z)
   $$

7. **Final expression**: Putting it all together, we have:
   $$
   \frac{1}{3} \left( \log(x) + 4 \log(y) - 2 \log(z) \right)
   $$

Thus, the expanded expression is:
$$
\frac{1}{3} \log(x) + \frac{4}{3} \log(y) - \frac{2}{3} \log(z)
$$

So, the final answer is:
$$
\log \sqrt[3]{\frac{x y^{4}}{z^{2}}} = \frac{1}{3} \log(x) + \frac{4}{3} \log(y) - \frac{2}{3} \log(z)
$$
$$ \log \sqrt[3]{\frac{x y^{4}}{z^{2}}} = \frac{1}{3} \log(x) + \frac{4}{3} \log(y) - \frac{2}{3} \log(z) $$

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