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To fully expand the given logarithmic expression using the properties of logarithms, let's break it down step-by-step.

### Expression:
\[
\log \left( \frac{x^3 \sqrt{z^5}}{y^4} \right)
\]

### Step 1: Apply the logarithm of a quotient rule
Using the property:
\[
\log \left( \frac{A}{B} \right) = \log A - \log B
\]
we get:
\[
\log \left( \frac{x^3 \sqrt{z^5}}{y^4} \right) = \log \left( x^3 \sqrt{z^5} \right) - \log \left( y^4 \right)
\]

### Step 2: Expand the first logarithm (numerator) using the logarithm of a product
Using the property:
\[
\log(AB) = \log A + \log B
\]
we can expand:
\[
\log \left( x^3 \sqrt{z^5} \right) = \log (x^3) + \log \left( \sqrt{z^5} \right)
\]

### Step 3: Simplify using the logarithm of a power rule
- For \( \log(x^3) \), apply:
\[
\log(x^n) = n \log(x)
\]
Thus:
\[
\log(x^3) = 3 \log x
\]

- For \( \log \left( \sqrt{z^5} \right) \), rewrite \( \sqrt{z^5} \) as \( z^{5/2} \). Then apply the power rule:
\[
\log \left( z^{5/2} \right) = \frac{5}{2} \log z
\]

### Step 4: Expand the second logarithm (denominator)
- For \( \log(y^4) \), apply the power rule:
\[
\log(y^4) = 4 \log y
\]

### Final Expression:
Now, putting all the pieces together:
\[
\log \left( \frac{x^3 \sqrt{z^5}}{y^4} \right) = 3 \log x + \frac{5}{2} \log z - 4 \log y
\]

Thus, the fully expanded logarithmic expression is:
\[
3 \log x + \frac{5}{2} \log z - 4 \log y
\]

Would you like further details on any part of this explanation? Here are a few related questions for practice:

1. How would you expand \(\log \left( \frac{a^2 b}{c^3} \right)\)?
2. Can you express \(\log \left( x^4 y^5 \right)\) using properties of logarithms?
3. What is the simplified form of \(\log \left( \sqrt{a^3 b} \right)\)?
4. How would you expand \(\log \left( \frac{x^2 y}{\sqrt{z}} \right)\)?
5. What is the general rule for expanding logarithms of quotients?

**Tip:** When expanding logarithms, always apply the quotient rule first, followed by the product and power rules!

Expand the logarithm fully using the properties of logs. Express the final answer in terms of log x, log y, and log z: log((x^3 * z^5)/(y^4))

Learn how to fully expand the logarithmic expression log((x^3 * sqrt(z^5))/(y^4)) using the properties of logarithms. This problem involves quotient, product, and power rules of logarithms, suitable for students in Grades 9-12.

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