QuestionCondense the following into a single log. $$4 \log _{3}(x)-\frac{1}{2} \log _{3}(y)-3 \log _{3}(2)=$$ $\square$

Studdy Solution
Combine the logarithms using the properties of logarithms. Specifically, use the product rule $\log_b(m) + \log_b(n) = \log_b(m \cdot n)$ and the quotient rule $\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)$.
Combine all terms into a single logarithm: $$\log_{3}(x^4) + \log_{3}(y^{-\frac{1}{2}}) + \log_{3}(2^{-3}) = \log_{3}\left(\frac{x^4}{y^{\frac{1}{2}} \cdot 2^3}\right)$$
Simplify the expression inside the logarithm: $$\log_{3}\left(\frac{x^4}{y^{\frac{1}{2}} \cdot 8}\right)$$
The condensed single logarithm is: $$\boxed{\log_{3}\left(\frac{x^4}{8\sqrt{y}}\right)}$$