Use logarithms to the base 10 to find the indicated logarithm. \( \log _{9} 11 \) \( \log _{9} 11=\square \) (Simplify your answer. Type an integer or a decimal rounded to four decimal places as needed.)

To find \( \log_{9} 11 \), we can use the change of base formula, which states: \[ \log_{b} a = \frac{\log_{k} a}{\log_{k} b} \] For this problem, we'll use base 10: \[ \log_{9} 11 = \frac{\log_{10} 11}{\log_{10} 9} \] First, we need to calculate \( \log_{10} 11 \) and \( \log_{10} 9 \): - \( \log_{10} 11 \approx 1.0414 \) - \( \log_{10} 9 = \log_{10} (3^2) = 2 \cdot \log_{10} 3 \approx 2 \cdot 0.4771 = 0.9542 \) Now, substituting these values into the formula: \[ \log_{9} 11 = \frac{1.0414}{0.9542} \approx 1.0934 \] Thus, \[ \log_{9} 11 \approx 1.0934 \]