Cross Crawford
01/10/2024 · Elementary School
Use the change-of-base formula and a calculator to evaluate the logarithm. Round your answer to three decimal places. \( \log _{1 / 6} 8 \) \( \log _{1 / 6} 8 \approx \square \) (Do not round until the final answer. Then round to three decimal places as needed.)
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Step-by-step Solution
To evaluate \( \log_{1/6} 8 \) 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 10 or \( e \)). Here, we will use base 10 for convenience.
Applying the change-of-base formula:
\[
\log_{1/6} 8 = \frac{\log_{10} 8}{\log_{10} (1/6)}
\]
Now, we can calculate \( \log_{10} 8 \) and \( \log_{10} (1/6) \) using a calculator.
1. Calculate \( \log_{10} 8 \):
\[
\log_{10} 8 \approx 0.903
\]
2. Calculate \( \log_{10} (1/6) \):
\[
\log_{10} (1/6) = \log_{10} 1 - \log_{10} 6 = 0 - \log_{10} 6 \approx -0.778
\]
Now, substituting these values into the change-of-base formula:
\[
\log_{1/6} 8 \approx \frac{0.903}{-0.778}
\]
Calculating this gives:
\[
\log_{1/6} 8 \approx -1.158
\]
Finally, rounding to three decimal places:
\[
\log_{1/6} 8 \approx -1.158
\]
Thus, the final answer is:
\[
\log_{1/6} 8 \approx -1.158
\]
Quick Answer
\[
\log_{1/6} 8 \approx -1.158
\]
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