To determine which of the given expressions is equivalent to [tex]\(\log \left(\frac{12}{5}\right)\)[/tex], we can use the properties of logarithms. Specifically, one key property is the logarithmic property of division:
[tex]\[
\log \left( \frac{a}{b} \right) = \log (a) - \log (b)
\][/tex]
Applying this property to our problem:
[tex]\[
\log \left( \frac{12}{5} \right) = \log (12) - \log (5)
\][/tex]
So, option A, [tex]\(\log (12) - \log (5)\)[/tex], is indeed equivalent to [tex]\(\log \left( \frac{12}{5} \right)\)[/tex].
Let's quickly review why the other options are not correct:
B. [tex]\(\log (12) + \log (5)\)[/tex] — This expression uses the logarithmic property for multiplication:
[tex]\[
\log (a \cdot b) = \log (a) + \log (b)
\][/tex]
So, [tex]\(\log (12) + \log (5)\)[/tex] would actually be [tex]\(\log (12 \cdot 5)\)[/tex], not [tex]\(\log \left(\frac{12}{5}\right)\)[/tex].
C. [tex]\(\log (12) \cdot \log (5)\)[/tex] — This expression does not correspond to any standard logarithmic property and is incorrect.
D. [tex]\(12 \cdot \log (5)\)[/tex] — This expression implies multiplying 12 by the logarithm of 5, which doesn't relate directly to [tex]\(\log \left(\frac{12}{5}\right)\)[/tex].
Therefore, the correct answer is:
[tex]\[
\text{A. } \log (12) - \log (5)
\][/tex]
