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

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.