https://file.aiquickdraw.com/mathbot/file/1728744533751gjplgi2n.jpg

The expression in the image is:

\[
\log \left( \frac{\sqrt{x^3} \cdot \sqrt[3]{y^2}}{z^4} \right)
\]

We can simplify this logarithmic expression using logarithmic properties. Let’s break it down step by step:

### Step 1: Simplify the radicals
\[
\sqrt{x^3} = x^{\frac{3}{2}}, \quad \text{and} \quad \sqrt[3]{y^2} = y^{\frac{2}{3}}
\]

Thus, the expression becomes:
\[
\log \left( \frac{x^{\frac{3}{2}} \cdot y^{\frac{2}{3}}}{z^4} \right)
\]

### Step 2: Apply the logarithmic property for division
\[
\log \left( \frac{A}{B} \right) = \log(A) - \log(B)
\]

This gives:
\[
\log \left( x^{\frac{3}{2}} \cdot y^{\frac{2}{3}} \right) - \log(z^4)
\]

### Step 3: Apply the logarithmic property for multiplication
\[
\log(AB) = \log(A) + \log(B)
\]

This gives:
\[
\log(x^{\frac{3}{2}}) + \log(y^{\frac{2}{3}}) - \log(z^4)
\]

### Step 4: Apply the logarithmic power rule
\[
\log(A^n) = n \log(A)
\]

Thus:
\[
\frac{3}{2} \log(x) + \frac{2}{3} \log(y) - 4 \log(z)
\]

This is the simplified form of the given logarithmic expression.

---

Would you like further clarification on any step, or do you have other questions? 

### Related Questions:
1. How can we apply these logarithmic properties to more complex expressions?
2. What are the properties of logarithms that are most commonly used?
3. How can the logarithmic power rule help in solving exponential equations?
4. What happens if the logarithmic base is not specified in expressions?
5. How does the change of base formula for logarithms work?

### Tip:
When simplifying logarithmic expressions, always try to apply the power rule and product rule to break down complex terms into manageable parts.

Simplify the logarithmic expression log(√(x³)√[³](y²) / z⁴).

This solution walks through the process of simplifying the logarithmic expression log(√(x³)√[³](y²) / z⁴) using key logarithmic rules. Learn to apply properties of logarithms, handle radicals, and express exponents. Suitable for high school students in Grades 10-12.

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