Suppose \( \log _{b}(x)=6, \log _{b}(y)=2 \), and \( \log _{b}(z)=8 \) Find the value of the following expression. \( \log _{b}\left(\frac{x^{3} y}{z^{2}}\right) \)

To find the value of \( \log _{b}\left(\frac{x^{3} y}{z^{2}}\right) \), we can use the properties of logarithms. First, we can rewrite the expression using the properties of logarithms: \[ \log_{b}\left(\frac{x^{3} y}{z^{2}}\right) = \log_{b}(x^{3}) + \log_{b}(y) - \log_{b}(z^{2}) \] Using the power rule of logarithms, this becomes: \[ \log_{b}(x^{3}) = 3 \log_{b}(x), \quad \log_{b}(z^{2}) = 2 \log_{b}(z) \] Substituting in the values we have: - \( \log_{b}(x) = 6 \) - \( \log_{b}(y) = 2 \) - \( \log_{b}(z) = 8 \) Now we substitute these values: \[ \log_{b}\left(\frac{x^{3} y}{z^{2}}\right) = 3 \cdot 6 + 2 - 2 \cdot 8 \] This simplifies to: \[ = 18 + 2 - 16 \] \[ = 18 + 2 - 16 = 4 \] So, the value of \( \log_{b}\left(\frac{x^{3} y}{z^{2}}\right) \) is \( \boxed{4} \).