Evaluate \( \log _{3} 27+2 \log _{3} 9-\log _{3} 243 \)

To evaluate the expression \( \log_{3} 27 + 2 \log_{3} 9 - \log_{3} 243 \), we can first simplify each logarithm. 1. We know that \( 27 = 3^3 \), so: \[ \log_{3} 27 = \log_{3} (3^3) = 3 \] 2. Next, \( 9 = 3^2 \), so: \[ \log_{3} 9 = \log_{3} (3^2) = 2 \] Thus, we have: \[ 2 \log_{3} 9 = 2 \times 2 = 4 \] 3. For \( 243 \), we note that \( 243 = 3^5 \), which gives us: \[ \log_{3} 243 = \log_{3} (3^5) = 5 \] Now we can substitute these values back into the original expression: \[ \log_{3} 27 + 2 \log_{3} 9 - \log_{3} 243 = 3 + 4 - 5 \] Calculating this: \[ 3 + 4 = 7 \\ 7 - 5 = 2 \] Therefore, the final answer is: \[ \boxed{2} \]