\begin{tabular}{l} e formula to evaluate the logarithm. \\ \( 2 \cdot \log _{3} 8-4 \cdot \log _{3} 2 \) \\ \hline\end{tabular}

To evaluate the expression \( 2 \cdot \log_{3} 8 - 4 \cdot \log_{3} 2 \), we can simplify using the properties of logarithms. First, we can express \( 8 \) as \( 2^3 \), which lets us rewrite \( \log_{3} 8 \) as \( \log_{3} (2^3) = 3 \cdot \log_{3} 2 \). Plugging that back into the expression gives us: \[ 2 \cdot (3 \cdot \log_{3} 2) - 4 \cdot \log_{3} 2 = 6 \cdot \log_{3} 2 - 4 \cdot \log_{3} 2 = 2 \cdot \log_{3} 2. \] So, the simplified result is \( 2 \cdot \log_{3} 2 \). Now, if you wanted to evaluate \( \log_{3} 2 \) further, you could use the change of base formula: \[ \log_{3} 2 = \frac{\log_{10} 2}{\log_{10} 3}, \] or you can also use natural logarithms: \[ \log_{3} 2 = \frac{\ln 2}{\ln 3}. \] This little change helps if you want a numerical approximation!