To determine which expression is equivalent to [tex]\(\log (3 \cdot 8)\)[/tex], we should use the properties of logarithms. Specifically, one of the most useful properties is the product rule, which states:
[tex]\[
\log(a \cdot b) = \log(a) + \log(b)
\][/tex]
Given the expression [tex]\(\log (3 \cdot 8)\)[/tex]:
1. Identify the components in the product inside the logarithm: In this case, [tex]\(a = 3\)[/tex] and [tex]\(b = 8\)[/tex].
2. Apply the product rule of logarithms:
[tex]\[
\log(3 \cdot 8) = \log(3) + \log(8)
\][/tex]
Checking the options given:
- Option A: [tex]\(3 \cdot \log(8)\)[/tex] - This expression is not correct because it multiplies [tex]\(\log(8)\)[/tex] by 3, which is not related to the product rule of logarithms.
- Option B: [tex]\(\log(3) \cdot \log(8)\)[/tex] - This expression is also incorrect because it multiplies [tex]\(\log(3)\)[/tex] by [tex]\(\log(8)\)[/tex], and the product rule involves addition, not multiplication.
- Option C: [tex]\(\log(3) + \log(8)\)[/tex] - This expression is correct based on the product rule.
- Option D: [tex]\(\log(3) - \log(8)\)[/tex] - This expression is incorrect as it uses subtraction, which corresponds to the quotient rule, not the product rule.
Thus, the correct expression equivalent to [tex]\(\log (3 \cdot 8)\)[/tex] is:
[tex]\[
\log(3) + \log(8)
\][/tex]
Therefore, the correct multiple choice answer is:
C. [tex]\(\log(3) + \log(8)\)[/tex]
