What is the value of log open parentheses fraction numerator x y cubed over denominator z squared end fraction close parentheses when given the following:

log(x) = 2

log(y) = - 4

log(z) = 3

Given the logarithmic identities and the provided values, we can solve for the expression:

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

### Step 1: Apply the logarithm properties
The properties of logarithms that will be useful are:

1. \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\)
2. \(\log(a \cdot b) = \log(a) + \log(b)\)
3. \(\log(a^n) = n \cdot \log(a)\)

Using these properties, we can break down the given expression:

\[
\log\left(\frac{x y^3}{z^2}\right) = \log(x) + \log(y^3) - \log(z^2)
\]

### Step 2: Simplify using the provided values
Now, substitute the provided values:

- \(\log(x) = 2\)
- \(\log(y) = -4\)
- \(\log(z) = 3\)

So,

\[
\log(y^3) = 3 \cdot \log(y) = 3 \cdot (-4) = -12
\]

\[
\log(z^2) = 2 \cdot \log(z) = 2 \cdot 3 = 6
\]

Substituting these into the original expression:

\[
\log\left(\frac{x y^3}{z^2}\right) = 2 + (-12) - 6
\]

### Step 3: Calculate the final value

\[
\log\left(\frac{x y^3}{z^2}\right) = 2 - 12 - 6 = -16
\]

Thus, the value of \(\log\left(\frac{x y^3}{z^2}\right)\) is \(\boxed{-16}\).

Would you like any additional details or explanations on this solution?

### Relative Questions
1. How do logarithmic properties help in simplifying complex expressions?
2. What are the key differences between logarithmic and exponential functions?
3. How would the result change if \(\log(y)\) was positive?
4. What is the significance of the base in logarithmic calculations?
5. Can you explain how to derive the logarithm of a quotient from the logarithms of its numerator and denominator?

### Tip
Always remember to apply the logarithm properties step-by-step, as they help break down complex expressions into simpler parts.

Explore how to calculate the logarithm of a complex expression using given log values. Solve the expression log((xy^3)/(z^2)) step-by-step with detailed explanations and computations. Suitable for advanced high school and college-level students.

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