To evaluate the logarithm [tex]\(\log_{\sqrt{5}} 6\)[/tex] using the change-of-base formula, we proceed as follows:
1. Step 1: Use the change-of-base formula to rewrite the given expression in terms of natural logarithms (or common logarithms):
[tex]\[
\log_{\sqrt{5}} 6 = \frac{\log 6}{\log \sqrt{5}}
\][/tex]
2. Step 2: Evaluate each logarithmic expression using a calculator.
- Evaluate [tex]\(\log 6\)[/tex]:
[tex]\[
\log 6 = 1.791759469228055
\][/tex]
- Evaluate [tex]\(\log \sqrt{5}\)[/tex]:
[tex]\[
\log \sqrt{5} = 0.8047189562170503
\][/tex]
3. Step 3: Substitute these values back into the change-of-base formula and perform the division:
[tex]\[
\log_{\sqrt{5}} 6 = \frac{1.791759469228055}{0.8047189562170503} = 2.2265655051187565
\][/tex]
4. Step 4: Round the result to three decimal places as needed:
[tex]\[
\log_{\sqrt{5}} 6 \approx 2.227
\][/tex]
Therefore, the final evaluated expression is:
[tex]\[
\log_{\sqrt{5}} 6 \approx 2.227
\][/tex]
This concludes the step-by-step solution using the change-of-base formula and evaluating using a calculator.
