Sure! Let's break down the given expression step by step to condense it into a single logarithm.
We start with the expression:
[tex]\[
\log 6 - 3 \log \left( \frac{1}{3} \right)
\][/tex]
Step 1: Apply the logarithm power rule.
The power rule for logarithms states that [tex]\( a \log b = \log(b^a) \)[/tex]. Applying this rule here, we get:
[tex]\[
3 \log \left( \frac{1}{3} \right) = \log \left( \left( \frac{1}{3} \right)^3 \right)
\][/tex]
Step 2: Simplify the term inside the logarithm.
Calculate [tex]\(\left( \frac{1}{3} \right)^3\)[/tex]:
[tex]\[
\left( \frac{1}{3} \right)^3 = \frac{1}{27}
\][/tex]
So, the expression now becomes:
[tex]\[
\log 6 - \log \left( \frac{1}{27} \right)
\][/tex]
Step 3: Apply the logarithm subtraction rule.
The subtraction rule for logarithms states that [tex]\( \log a - \log b = \log \left( \frac{a}{b} \right) \)[/tex]. Using this rule:
[tex]\[
\log 6 - \log \left( \frac{1}{27} \right) = \log \left( \frac{6}{\frac{1}{27}} \right)
\][/tex]
Step 4: Simplify the fraction inside the logarithm.
[tex]\[
\frac{6}{\frac{1}{27}} = 6 \times 27 = 162
\][/tex]
So, the expression simplifies to:
[tex]\[
\log 162
\][/tex]
Therefore, the single logarithm that condenses the original expression is:
[tex]\[
\log 162
\][/tex]
