What is the value of \( \log \left(\frac{x^{2} y^{3}}{z}\right) \) when given the following: \( \begin{array}{l}\log (x)=3 \\ \log (y)=2 \\ \log (z)=-1 \\ \text { a.) } 12 \\ \text { b.) } 16 \\ \text { c.) } 13 \\ \text { d.) } 18\end{array} \)

To evaluate \( \log \left(\frac{x^{2} y^{3}}{z}\right) \), we can use the properties of logarithms. First, we can break it down as follows: \[ \log \left(\frac{x^{2} y^{3}}{z}\right) = \log \left(x^{2} y^{3}\right) - \log(z) \] Next, we can further break down \( \log \left(x^{2} y^{3}\right) \): \[ \log \left(x^{2} y^{3}\right) = \log(x^{2}) + \log(y^{3}) = 2\log(x) + 3\log(y) \] Now, substituting the given values: - \( \log(x) = 3 \) - \( \log(y) = 2 \) - \( \log(z) = -1 \) We can substitute these values into the equation: \[ \log \left(x^{2} y^{3}\right) = 2(3) + 3(2) = 6 + 6 = 12 \] Now, substituting back into the original equation: \[ \log \left(\frac{x^{2} y^{3}}{z}\right) = 12 - (-1) = 12 + 1 = 13 \] Thus, the value of \( \log \left(\frac{x^{2} y^{3}}{z}\right) \) is \( \boxed{13} \).