To expand the given logarithmic expression:

$\log \left( \frac{x}{\sqrt[3]{z^5 y^2}} \right)$

we will use the properties of logarithms, such as:

1. The quotient rule: $\log \left( \frac{a}{b} \right) = \log(a) - \log(b)$
2. The product rule: $\log(a \cdot b) = \log(a) + \log(b)$
3. The power rule: $\log(a^b) = b \log(a)$
4. The root rule: $\log \left( \sqrt[n]{a} \right) = \frac{1}{n} \log(a)$

Step-by-step expansion:

1. Apply the quotient rule:

[ \log \left( \frac{x}{\sqrt[3]{z^5 y^2}} \right) = \log(x) - \log \left( \sqrt[3]{z^5 y^2} \right) ]

2. Simplify the second term using the root rule:

$\log \left( \sqrt[3]{z^5 y^2} \right) = \frac{1}{3} \log \left( z^5 y^2 \right)$

3. Apply the product rule to the logarithm of the product inside:

$\log \left( z^5 y^2 \right) = \log(z^5) + \log(y^2)$

4. Apply the power rule to both terms inside the logarithms:

$\log(z^5) = 5 \log(z), \quad \log(y^2) = 2 \log(y)$

Thus, we have:

$\log \left( \frac{x}{\sqrt[3]{z^5 y^2}} \right) = \log(x) - \frac{1}{3} \left( 5 \log(z) + 2 \log(y) \right)$

5. Distribute the $\frac{1}{3}$:

$= \log(x) - \frac{5}{3} \log(z) - \frac{2}{3} \log(y)$

Final expanded form:

$\log(x) - \frac{5}{3} \log(z) - \frac{2}{3} \log(y)$

This is the fully expanded form of the logarithmic expression.

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Would you like more details or have any questions about the steps? Here are some related questions you might find interesting:

1. How do logarithmic identities apply to solving equations?
2. Can the power rule be used with other types of roots, like square roots or fourth roots?
3. How does the change of base formula help when calculating logarithms?
4. What are some common mistakes when expanding logarithmic expressions?
5. How would you simplify a logarithmic expression involving multiple bases?

Tip: Always break down complex expressions step by step using the basic logarithmic rules. This makes the process clearer and more manageable!