Suppose \( \log _{\alpha} x=5, \log _{a} y=3 \), and \( \log _{a} z=-1 \) Find the value of the following expression. \( \log _{a}\left(\frac{x^{2} y}{z^{3}}\right) \) 16 13 19

To solve for \( \log_{a}\left(\frac{x^{2} y}{z^{3}}\right) \), we can use the properties of logarithms: 1. Using the product rule: \( \log_{a}(mn) = \log_{a} m + \log_{a} n \) 2. Using the quotient rule: \( \log_{a}\left(\frac{m}{n}\right) = \log_{a} m - \log_{a} n \) 3. Using the power rule: \( \log_{a}(m^n) = n \log_{a} m \) Given: - \( \log_{\alpha} x = 5 \) which means \( x = \alpha^5 \) - \( \log_{a} y = 3 \) which means \( y = a^3 \) - \( \log_{a} z = -1 \) which means \( z = a^{-1} \) We express \( \log_a\left(\frac{x^{2} y}{z^{3}}\right) \) as follows: \[ \log_{a}\left(\frac{x^{2} y}{z^{3}}\right) = \log_{a}(x^{2}) + \log_{a}(y) - \log_{a}(z^{3}) \] Applying the rules of logarithms: \[ = 2 \log_{a}(x) + \log_{a}(y) - 3 \log_{a}(z) \] Now we need to express \( \log_{a}(x) \): \[ \log_{a}(x) = \log_{a} (\alpha^{5}) = 5 \log_{a}(\alpha) \] Next, substitute the known values: \[ = 2(5 \log_{a}(\alpha)) + 3 - 3(-1) \] \[ = 10 \log_{a}(\alpha) + 3 + 3 \] \[ = 10 \log_{a}(\alpha) + 6 \] At this point, we need the value of \( \log_{a}(\alpha) \). However, since we want the overall expression, and the information to calculate \( \log_{a}(\alpha) \) isn't provided, we cannot determine the exact numeric answer at this point. However, if we assume, for instance, \( \log_a(\alpha) = 1 \): \[ \log_{a}\left(\frac{x^{2} y}{z^{3}}\right) = 10(1) + 6 = 16 \] Therefore, the value of the expression could be \( 16 \) depending on the assumption made about \( \log_{a}(\alpha) \). Final answer: 16