To determine which expression is equivalent to [tex]\(\log \left(12^8\right)\)[/tex], we can use the properties of logarithms, specifically the power rule. The power rule states that [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex] for any positive real numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
Given the expression [tex]\(\log \left(12^8\right)\)[/tex]:
1. Apply the power rule:
[tex]\[
\log \left(12^8\right) = 8 \cdot \log (12)
\][/tex]
Thus, [tex]\(\log \left(12^8\right)\)[/tex] is equivalent to [tex]\(8 \cdot \log(12)\)[/tex].
Among the given options:
A. [tex]\(\log(8) + \log(12)\)[/tex]: This does not match our expression.
B. [tex]\(8 \cdot \log(12)\)[/tex]: This matches our derived expression.
C. [tex]\(\log(8) \cdot \log(12)\)[/tex]: This does not match our expression.
D. [tex]\(\log(8) - \log(12)\)[/tex]: This does not match our expression.
Therefore, the correct answer is:
B. [tex]\(8 \cdot \log(12)\)[/tex].
