To expand the given logarithmic expression, we can apply the properties of logarithms systematically. The original expression is:

$\log \left( \frac{x}{y^4 \sqrt[3]{z}} \right)$

Step-by-Step Expansion

1. Apply the Quotient Rule:
   The quotient rule for logarithms states that $\log \left( \frac{a}{b} \right) = \log a - \log b$. Applying this, we get:

   [ \log \left( \frac{x}{y^4 \sqrt[3]{z}} \right) = \log x - \log \left( y^4 \sqrt[3]{z} \right) ]

2. Apply the Product Rule:
   Now, apply the product rule inside the second logarithm. The product rule states that $\log(ab) = \log a + \log b$:

   $\log x - \left( \log(y^4) + \log\left(\sqrt[3]{z}\right) \right)$

   Simplifying, we get:

   $\log x - \log(y^4) - \log\left(\sqrt[3]{z}\right)$

3. Apply the Power Rule:
   Now, apply the power rule for logarithms, which states $\log(a^b) = b \cdot \log a$:

   • For $\log(y^4)$:
     $\log(y^4) = 4 \cdot \log y$

   • For $\log\left(\sqrt[3]{z}\right)$:
     Recall that $\sqrt[3]{z} = z^{1/3}$, so $\log\left(\sqrt[3]{z}\right) = \frac{1}{3} \cdot \log z$

   Substituting these into the expression, we get:

   $\log x - 4 \cdot \log y - \frac{1}{3} \cdot \log z$

Final Answer

So, the fully expanded expression is:

$\log x - 4 \log y - \frac{1}{3} \log z$

Would you like more details on any of the steps or have any questions?

Related Questions

1. How would you expand $\log \left( \frac{a b}{c^2 d^{1/2}} \right)$?
2. What is the expanded form of $\log \left( x^3 y z^4 \right)$?
3. How do the properties of logarithms simplify expressions involving products and quotients?
4. Why does the power rule work for logarithms, and how is it derived?
5. How would you simplify $\log(x^2) - \log(x^3)$?

Tip

Remember, breaking down expressions step-by-step with logarithmic properties like quotient, product, and power rules can make complex expressions easier to handle.