Mcguire Todd
01/10/2024 · Junior High School
Write the expression as a single logarithm. Express powers as factors. \( \log _{3} \sqrt{x}-\log _{3} x^{3} \) \( \log _{3} \sqrt{x}-\log _{3} x^{3}=\square \) (Type an exact answer. Use integers or fractions for any numbers in the expression.)
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Step-by-step Solution
Calculate or simplify the expression \( \log_{3} \sqrt{x} - \log_{3} x^{3} \).
Simplify the expression by following steps:
- step0: Solution:
\(\log_{3}{\left(\sqrt{x}\right)}-\log_{3}{\left(x^{3}\right)}\)
- step1: Transform the expression:
\(\log_{3}{\left(\frac{\sqrt{x}}{x^{3}}\right)}\)
- step2: Rearrange the terms:
\(\log_{3}{\left(x^{-\frac{5}{2}}\right)}\)
- step3: Simplify the expression:
\(-\frac{5}{2}\log_{3}{\left(x\right)}\)
The expression \( \log_{3} \sqrt{x} - \log_{3} x^{3} \) can be simplified to \( -\frac{5}{2} \log_{3} x \).
Quick Answer
\( -\frac{5}{2} \log_{3} x \)
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