What is the value of \( \log \left(\frac{x^{2} y^{3}}{z}\right) \) when given the following \( \log (x)=3 \) \( \log (y)=2 \) \( \log (z)=-1 \)

To find the value of \( \log \left(\frac{x^{2} y^{3}}{z}\right) \), we can use the properties of logarithms. Specifically, we first apply the quotient and product rules: \[ \log \left(\frac{x^{2} y^{3}}{z}\right) = \log (x^{2} y^{3}) - \log (z) \] Now, we can separate the logarithm of the product further: \[ \log (x^{2} y^{3}) = \log (x^{2}) + \log (y^{3}) = 2 \log (x) + 3 \log (y) \] We can now substitute the values we have: \[ \log (x) = 3, \quad \log (y) = 2, \quad \log (z) = -1 \] Substituting these values in gives: \[ \log (x^{2}) + \log (y^{3}) = 2(3) + 3(2) = 6 + 6 = 12 \] Now we include the logarithm of \( z \): \[ \log \left(\frac{x^{2} y^{3}}{z}\right) = 12 - (-1) = 12 + 1 = 13 \] Thus, the value of \( \log \left(\frac{x^{2} y^{3}}{z}\right) \) is \( \boxed{13} \).