To evaluate \( \log_{11} 22 \) using the change-of-base formula, we can use the formula:
\[ \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \]
where \( a = 11 \), \( b = 22 \), and \( c \) can be any positive number (commonly, we use 10 or \( e \)). Here, we will use base 10 logarithms.
Applying the change-of-base formula, we get:
\[ \log_{11} 22 = \frac{\log_{10} 22}{\log_{10} 11} \]
Next, we calculate \( \log_{10} 22 \) and \( \log_{10} 11 \) using a calculator:
\[ \log_{10} 22 \approx 1.3424 \] \[ \log_{10} 11 \approx 1.0414 \]
Now substitute these values into the formula:
\[ \log_{11} 22 = \frac{1.3424}{1.0414} \approx 1.2907 \]
Finally, rounding this to three decimal places, we get:
\[ \log_{11} 22 \approx 1.291 \]
Thus, the answer is:
\[ \boxed{1.291} \]