To determine which expressions are equivalent to [tex]\(\log 2 - \log 6\)[/tex], we need to simplify the given expression step-by-step and compare it to the options provided.
First, recall the logarithmic property:
[tex]\[
\log a - \log b = \log \left(\frac{a}{b}\right)
\][/tex]
Using this property, we can simplify [tex]\(\log 2 - \log 6\)[/tex]:
[tex]\[
\log 2 - \log 6 = \log \left(\frac{2}{6}\right)
\][/tex]
Next, simplify the fraction inside the logarithm:
[tex]\[
\log \left(\frac{2}{6}\right) = \log \left(\frac{1}{3}\right)
\][/tex]
So, [tex]\(\log 2 - \log 6\)[/tex] simplifies to [tex]\(\log \left(\frac{1}{3}\right)\)[/tex].
Now let's compare this result to the options given:
- Option A: [tex]\(\log 2\)[/tex]
[tex]\[
\log 2 \quad \text{(This is not equivalent to } \log \left(\frac{1}{3}\right) \text{)}
\][/tex]
- Option B: [tex]\(\log 3\)[/tex]
[tex]\[
\log 3 \quad \text{(This is not equivalent to } \log \left(\frac{1}{3}\right) \text{)}
\][/tex]
- Option C: [tex]\(\log \left(\frac{1}{3}\right)\)[/tex]
[tex]\[
\log \left(\frac{1}{3}\right) \quad \text{(This is exactly } \log \left(\frac{1}{3}\right) \text{)}
\][/tex]
- Option D: [tex]\(\log 2 + \log \left(\frac{1}{6}\right)\)[/tex]
Let's simplify [tex]\(\log 2 + \log \left(\frac{1}{6}\right)\)[/tex]:
[tex]\[
\log 2 + \log \left(\frac{1}{6}\right) = \log \left(2 \cdot \frac{1}{6}\right) = \log \left(\frac{2}{6}\right) = \log \left(\frac{1}{3}\right)
\][/tex]
This matches [tex]\(\log \left(\frac{1}{3}\right)\)[/tex].
Given the simplified result of [tex]\(\log 2 - \log 6\)[/tex], the equivalent expressions are:
- Option C: [tex]\(\log \left(\frac{1}{3}\right)\)[/tex]
- Option D: [tex]\(\log 2 + \log \left(\frac{1}{6}\right)\)[/tex]
