Question

Use a change of base formula to evaluate the logarithm. Give your answer rounded to four decimal places.
log Subscript 3 Baseline StartRoot 11 EndRoot
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Part 1
log Subscript 3 Baseline StartRoot 11 EndRoot equals
  
enter your response here
​(Simplify your answer. Do not round until the final answer. Then round to four decimal places as​ needed.)

Answer 1

To evaluate \( \log_3 \sqrt{11} \) using the change of base formula, we can use the following formula:

\[
\log_b a = \frac{\log_k a}{\log_k b}
\]

where \( k \) can be any positive number (commonly base 10 or \( e \)).

For this problem, let's apply the change of base formula:

\[
\log_3 \sqrt{11} = \frac{\log_{10} \sqrt{11}}{\log_{10} 3}
\]

First, we simplify \( \sqrt{11} \):

\[
\sqrt{11} = 11^{1/2}
\]

Thus,

\[
\log_{10} \sqrt{11} = \log_{10} (11^{1/2}) = \frac{1}{2} \log_{10} 11
\]

Putting it all together, we have:

\[
\log_3 \sqrt{11} = \frac{\frac{1}{2} \log_{10} 11}{\log_{10} 3} = \frac{\log_{10} 11}{2 \log_{10} 3}
\]

Now, we can calculate the logarithms using a calculator:

\[
\log_{10} 11 \approx 1.0414
\]
\[
\log_{10} 3 \approx 0.4771
\]

Now substitute these values into the expression:

\[
\log_3 \sqrt{11} \approx \frac{1.0414}{2 \times 0.4771} \approx \frac{1.0414}{0.9542} \approx 1.0925
\]

Thus, rounded to four decimal places, we get:

\[
\log_3 \sqrt{11} \approx 1.0925
\]

So, \( \log_3 \sqrt{11} \) equals approximately **1.0925**.