Which expressions are equivalent to the one below? Check all that apply.

[tex]\(\log 2 - \log 6\)[/tex]

A. [tex]\(\log 2\)[/tex]
B. [tex]\(\log 3\)[/tex]
C. [tex]\(\log \left(\frac{1}{3}\right)\)[/tex]
D. [tex]\(\log (2) + \log \left(\frac{1}{6}\right)\)[/tex]



Answer :

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]