Step 1 :Apply the property of logarithms \(\log_b(M/N) = \log_b(M) - \log_b(N)\) to the given expression: \(\log _{2}(8x) - \log _{2}(\sqrt[3]{y})\)
Step 2 :Apply the property of logarithms \(\log_b(MN) = \log_b(M) + \log_b(N)\) to the first term: \(\log _{2}(8) + \log _{2}(x) - \log _{2}(\sqrt[3]{y})\)
Step 3 :Apply the property of logarithms \(\log_b(M^p) = p \cdot \log_b(M)\) to the first and last term: \(3 \cdot \log _{2}(2) + \log _{2}(x) - \frac{1}{3} \cdot \log _{2}(y)\)
Step 4 :Simplify the expression, since \(\log _{2}(2) = 1\), the expression simplifies to: \(3 + \log _{2}(x) - \frac{1}{3} \cdot \log _{2}(y)\)
Step 5 :So, the expanded form of the given logarithmic expression is \(\boxed{3 + \log _{2}(x) - \frac{1}{3} \cdot \log _{2}(y)}\)