To identify the original expression, we can use the change of base formula for logarithms, which states:
[tex]\[
\log_b a = \frac{\log a}{\log b}
\][/tex]
We are given the expression:
[tex]\[
\frac{\log \frac{1}{3}}{\log 2}
\][/tex]
We need to identify which of the given options matches this expression when the change of base formula is applied.
Let's consider each option one by one:
1. [tex]\(\log_{\frac{1}{3}} 2\)[/tex]
- Using the change of base formula:
[tex]\[
\log_{\frac{1}{3}} 2 = \frac{\log 2}{\log \frac{1}{3}}
\][/tex]
- This does not match the given expression [tex]\(\frac{\log \frac{1}{3}}{\log 2}\)[/tex].
2. [tex]\(\log_{\frac{1}{2}} 3\)[/tex]
- Using the change of base formula:
[tex]\[
\log_{\frac{1}{2}} 3 = \frac{\log 3}{\log \frac{1}{2}}
\][/tex]
- This does not match the given expression [tex]\(\frac{\log \frac{1}{3}}{\log 2}\)[/tex].
3. [tex]\(\log_2 \frac{1}{3}\)[/tex]
- Using the change of base formula:
[tex]\[
\log_2 \frac{1}{3} = \frac{\log \frac{1}{3}}{\log 2}
\][/tex]
- This exactly matches the given expression [tex]\(\frac{\log \frac{1}{3}}{\log 2}\)[/tex].
4. [tex]\(\log_5 \frac{1}{2}\)[/tex]
- Using the change of base formula:
[tex]\[
\log_5 \frac{1}{2} = \frac{\log \frac{1}{2}}{\log 5}
\][/tex]
- This does not match the given expression [tex]\(\frac{\log \frac{1}{3}}{\log 2}\)[/tex].
Therefore, the original expression that matches the given expression [tex]\(\frac{\log \frac{1}{3}}{\log 2}\)[/tex] is:
[tex]\[
\boxed{\log_2 \frac{1}{3}}
\][/tex]
