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The given expression is:

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

We need to express this as a sum and difference of logarithms with no exponents on the argument. Here's the step-by-step breakdown:

### Step 1: Use the logarithmic property of division
\[
\ln\left(\frac{A}{B}\right) = \ln(A) - \ln(B)
\]
Here, \( A = x^5 y^2 \) and \( B = z^3 \), so:
\[
\ln\left(\frac{x^5 y^2}{z^3}\right) = \ln(x^5 y^2) - \ln(z^3)
\]

### Step 2: Use the logarithmic property of multiplication
\[
\ln(AB) = \ln(A) + \ln(B)
\]
For \( A = x^5 \) and \( B = y^2 \), this gives:
\[
\ln(x^5 y^2) = \ln(x^5) + \ln(y^2)
\]

### Step 3: Use the logarithmic property of exponents
\[
\ln(A^n) = n \ln(A)
\]
Applying this to \( x^5 \), \( y^2 \), and \( z^3 \):
\[
\ln(x^5) = 5 \ln(x), \quad \ln(y^2) = 2 \ln(y), \quad \ln(z^3) = 3 \ln(z)
\]

### Step 4: Combine everything
Now, substituting back into the expression:
\[
\ln\left(\frac{x^5 y^2}{z^3}\right) = 5 \ln(x) + 2 \ln(y) - 3 \ln(z)
\]

This matches the final option in the list:

**Answer: \( 5 \ln(x) + 2 \ln(y) - 3 \ln(z) \)**

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Would you like more details or have any questions about the process?

Here are 5 related questions to consider:
1. What other logarithmic properties are essential when simplifying expressions like this?
2. How would the solution change if the base of the logarithms was different, say \( \log_{10} \) instead of \( \ln \)?
3. Can the logarithmic properties be extended to handle expressions with radicals (e.g., square roots)?
4. How would you differentiate the expression \( 5 \ln(x) + 2 \ln(y) - 3 \ln(z) \) with respect to \( x \), \( y \), or \( z \)?
5. What happens if one of the variables \( x \), \( y \), or \( z \) approaches zero in the original logarithmic function?

**Tip**: When simplifying logarithmic expressions, always look for opportunities to apply the properties of exponents, products, and quotients to break down complex arguments.

Write the expression ln((x^5 * y^2) / z^3) as a sum and difference of logarithms with no exponents on the argument.

Learn how to simplify the logarithmic expression ln((x^5 * y^2) / z^3) using properties of logarithms. Follow a step-by-step breakdown covering key logarithmic rules, including division, multiplication, and exponent properties. Suitable for students in Grades 10-12.

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