Question: What is the value of $\log\left( \frac{x^{2}y^{3}}{z} \right)$ when given the following:
$\log(x) = 3$
$\log(y) = 2$
$\log(z) = -1$
a.) 18
b.) 12
c.) 16
d.) 13
1. **State the problem:** We need to find the value of $\log\left( \frac{x^{2}y^{3}}{z} \right)$ given $\log(x) = 3$, $\log(y) = 2$, and $\log(z) = -1$.
2. **Recall the properties of logarithms:**
- $\log(ab) = \log a + \log b$
- $\log\left( \frac{a}{b} \right) = \log a - \log b$
- $\log(a^{n}) = n \log a$
3. **Apply the properties:**
\[
\log\left( \frac{x^{2}y^{3}}{z} \right) = \log(x^{2}y^{3}) - \log(z) = \log(x^{2}) + \log(y^{3}) - \log(z)
\]
4. **Use the power rule:**
\[
= 2\log(x) + 3\log(y) - \log(z)
\]
5. **Substitute the given values:**
\[
= 2(3) + 3(2) - (-1) = 6 + 6 + 1
\]
6. **Calculate the sum:**
\[
6 + 6 + 1 = 13
\]
**Final answer:** $13$ (option d.)