Find the value of the following expression. \( \log _{a}\left(\frac{x^{2} y}{z^{3}}\right) \)

To find the value of the expression \( \log _{a}\left(\frac{x^{2} y}{z^{3}}\right) \), we can apply the properties of logarithms. Using the quotient rule for logarithms, we can break this down as: \[ \log _{a}\left(\frac{x^{2} y}{z^{3}}\right) = \log_{a}(x^{2}y) - \log_{a}(z^{3}) \] Next, we apply the product rule to \( \log_{a}(x^{2}y) \): \[ \log_{a}(x^{2}y) = \log_{a}(x^{2}) + \log_{a}(y) \] For \( \log_{a}(x^{2}) \), we can use the power rule: \[ \log_{a}(x^{2}) = 2\log_{a}(x) \] And for \( \log_{a}(z^{3}) \): \[ \log_{a}(z^{3}) = 3\log_{a}(z) \] Putting it all together, we have: \[ \log_{a}\left(\frac{x^{2} y}{z^{3}}\right) = (2\log_{a}(x) + \log_{a}(y)) - 3\log_{a}(z) \] So, we can simplify it to: \[ \log_{a}\left(\frac{x^{2} y}{z^{3}}\right) = 2\log_{a}(x) + \log_{a}(y) - 3\log_{a}(z) \] That's the value of the expression!