Pregunta
If \( \log _{3} 41 \approx 3.38 \) and \( \log _{3} 2 \approx 0.631 \), what is \( \log _{3} 82 \), to the nearest thousandth? (1 point) Check answer Remaining Attempts : 3
Ask by Alexander Goodwin. in the United States
Mar 11,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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\( \log_{3} 82 \approx 4.011 \)
Solución
We can express \(82\) as the product of \(2\) and \(41\):
\[
82 = 2 \times 41
\]
Using the logarithm product rule:
\[
\log_{3} 82 = \log_{3} 2 + \log_{3} 41
\]
Substitute the given values:
\[
\log_{3} 82 \approx 0.631 + 3.38 = 4.011
\]
Thus, to the nearest thousandth,
\[
\log_{3} 82 \approx 4.011
\]
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Como
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Extra Insights
To find \( \log_{3} 82 \), we can express 82 as \( 2 \times 41 \). Using the properties of logarithms, we can calculate: \[ \log_{3} 82 = \log_{3} (2 \times 41) = \log_{3} 2 + \log_{3} 41 \] Now we can substitute the given approximate values: \[ \log_{3} 82 \approx \log_{3} 2 + \log_{3} 41 \approx 0.631 + 3.38 \] Adding these values together: \[ \log_{3} 82 \approx 0.631 + 3.38 = 4.011 \] Now, rounding to the nearest thousandth, we find: \[ \log_{3} 82 \approx 4.011 \] So the final answer is \( \boxed{4.011} \).

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