Question

Use the change-of-base formula to evaluate the expression. log_{11}22
(Do not round until the final answer. Then round to three decimal places as needed.)

Answer 1

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:

\[
\boxed{1.291}
\]