Question
Condense the expression. \[ \log 6-3 \log \frac{1}{3} \] \( \log [?] \)
Ask by Johnson Christensen. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\(\log 162\)
Solution
To condense the expression \(\log 6 - 3 \log \frac{1}{3}\) into the form \(\log [\,?\,]\), follow these steps using logarithm properties:
1. **Apply the Power Rule**:
\[
3 \log \frac{1}{3} = \log \left(\frac{1}{3}\right)^3 = \log \frac{1}{27}
\]
2. **Combine the Logarithms**:
\[
\log 6 - \log \frac{1}{27} = \log \left(\frac{6}{\frac{1}{27}}\right) = \log (6 \times 27)
\]
3. **Simplify the Expression**:
\[
6 \times 27 = 162
\]
Therefore, the condensed form of the expression is:
\[
\log 162
\]
Answer: \(\log 162\)
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Simplify this solution Beyond the Answer
To condense the expression \(\log 6 - 3 \log \frac{1}{3}\), we can use the properties of logarithms. Recall that \(a \log b = \log b^a\) and \(\log a - \log b = \log \frac{a}{b}\). First, rewrite \(3 \log \frac{1}{3}\) as \(\log \left(\frac{1}{3}\right)^3\): \[ 3 \log \frac{1}{3} = \log \left(\frac{1}{3}\right)^3 = \log \left(\frac{1}{27}\right). \] Now substitute this back into the original expression: \[ \log 6 - \log \left(\frac{1}{27}\right) = \log \left( \frac{6}{\frac{1}{27}} \right). \] This simplifies to: \[ \log \left( 6 \times 27 \right) = \log 162. \] So, the final condensed expression is: \[ \log[162]. \]
