The expression [tex]\(\frac{\log \frac{1}{3}}{\log 2}\)[/tex] is the result of applying the change of base formula to which original expression?

A. [tex]\(\log _{\frac{1}{3}} 2\)[/tex]
B. [tex]\(\log _{\frac{1}{2}} 3\)[/tex]
C. [tex]\(\log _2 \frac{1}{3}\)[/tex]
D. [tex]\(\log _5 \frac{1}{2}\)[/tex]



Answer :

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]