Problem
Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. \[ \log _{2}\left(\frac{x^{4} y^{3}}{7}\right) \] A. $\left(\log _{2} x\right)^{4}+\left(\log _{2} y\right)^{3}-\log _{2} 7$ B. $4 \log _{2} x+3 \log _{2} y-\log _{2} 7$ C. $4 \log _{2} x+3 \log _{2} y+\log _{2} 7$ D. $\left(4 \log _{2} x\right)\left(3 \log _{2} y\right) \div \log _{2} 7$
Solution
Step 1 ：Given the expression \(\log _{2}\left(\frac{x^{4} y^{3}}{7}\right)\)
Step 2 ：Using the properties of logarithms, we can rewrite the expression as \(\log _{2} x^{4} + \log _{2} y^{3} - \log _{2} 7\)
Step 3 ：Applying the property \(\log_b(m^n) = n\log_b(m)\), we can further simplify the expression to \(4 \log _{2} x + 3 \log _{2} y - \log _{2} 7\)
Step 4 ：Final Answer: \(\boxed{4 \log _{2} x + 3 \log _{2} y - \log _{2} 7}\)
