Evaluate \( \log _{9} 27 \) without a calculator. Note: If your answer is a fraction, enter your answer as a reduced improper fraction. Ex. \( 4 / 3 \) for \( \frac{4}{3} \). type your answer...

To evaluate \( \log_{9} 27 \), we can use the change of base formula. First, we express both 9 and 27 in terms of powers of 3: \[ 9 = 3^2 \quad \text{and} \quad 27 = 3^3. \] Now, substituting these into the logarithm: \[ \log_{9} 27 = \log_{3^2} 3^3. \] Using the change of base formula, we have: \[ \log_{3^2} 3^3 = \frac{\log_{3} 3^3}{\log_{3} 3^2}. \] Now, we can evaluate each logarithm: \[ \log_{3} 3^3 = 3 \quad \text{and} \quad \log_{3} 3^2 = 2. \] Substituting these values back into our fraction gives us: \[ \log_{9} 27 = \frac{3}{2}. \] Therefore, the answer is \[ \boxed{\frac{3}{2}}. \]