Question
\( \log _{3} 405-\log _{3} 5=\square \) (Type an exact answer in simplified form. Use integers or frac
Ask by Summers Lewis. in Saudi Arabia
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( \log_{3}(405) - \log_{3}(5) = \log_{3}\left(\frac{405}{5}\right) = \log_{3}(81) = 4 \)
Solution
Calculate or simplify the expression \( \log_{3}(405)-\log_{3}(5) \).
Calculate the value by following steps:
- step0: Calculate:
\(\log_{3}{\left(405\right)}-\log_{3}{\left(5\right)}\)
- step1: Use the logarithm product rule:
\(\log_{3}{\left(\frac{405}{5}\right)}\)
- step2: Divide the terms:
\(\log_{3}{\left(81\right)}\)
- step3: Write in exponential form:
\(\log_{3}{\left(3^{4}\right)}\)
- step4: Simplify the expression:
\(4\)
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Simplify this solution Beyond the Answer
To simplify \( \log_{3} 405 - \log_{3} 5 \), you can use the logarithm property that states \( \log_{a} b - \log_{a} c = \log_{a} \left(\frac{b}{c}\right) \). So, we have: \[ \log_{3} 405 - \log_{3} 5 = \log_{3} \left(\frac{405}{5}\right) \] Now, calculate \( \frac{405}{5} \): \[ \frac{405}{5} = 81 \] Now plug that back into the logarithm: \[ \log_{3} 81 \] Since \( 81 = 3^4 \), we can rewrite the logarithm: \[ \log_{3} 81 = \log_{3} (3^4) = 4 \] Thus, the simplified answer is: \[ \boxed{4} \]
