Let's find the equivalent expression for [tex]\(\log 5\)[/tex] by evaluating the given options one by one.
### Option A) [tex]\(\log 15 \div \log 3\)[/tex]
To check if this expression is equivalent to [tex]\(\log 5\)[/tex], we need to use the properties of logarithms. However, evaluating [tex]\(\log 15 / \log 3\)[/tex] would not simplify to [tex]\(\log 5\)[/tex]. This expression represents a different logarithmic ratio and is not equal to [tex]\(\log 5\)[/tex].
### Option B) [tex]\(\log 15 - \log 3\)[/tex]
Using the logarithm property that states [tex]\(\log(a) - \log(b) = \log(a/b)\)[/tex]:
[tex]\[
\log 15 - \log 3 = \log \left( \frac{15}{3} \right) = \log 5
\][/tex]
This option simplifies directly to [tex]\(\log 5\)[/tex].
### Option C) [tex]\(\frac{1}{2} \log 10\)[/tex]
Evaluating this expression:
[tex]\[
\frac{1}{2} \log 10 \neq \log 5
\][/tex]
This expression does not simplify to [tex]\(\log 5\)[/tex].
### Option D) [tex]\(\log 15 - \log 10\)[/tex]
Using the same property of logarithms:
[tex]\[
\log 15 - \log 10 = \log \left( \frac{15}{10} \right) = \log 1.5
\][/tex]
This expression simplifies to [tex]\(\log 1.5\)[/tex], which is not equal to [tex]\(\log 5\)[/tex].
### Option E) None of these
Since we found that Option B simplifies directly to [tex]\(\log 5\)[/tex], this option is incorrect.
Therefore, the correct answer is [tex]\(\boxed{\log 15 - \log 3}\)[/tex], which corresponds to Option B.
