To determine which of these expressions is equivalent to [tex]\(\log \left(4^6\right)\)[/tex], let's start by using the properties of logarithms.
The expression [tex]\(\log \left(4^6\right)\)[/tex] can be simplified using the power rule of logarithms, which states that [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex].
Given [tex]\(\log \left(4^6\right)\)[/tex], we apply the power rule:
[tex]\[
\log \left(4^6\right) = 6 \cdot \log (4)
\][/tex]
Therefore, the correct expression that is equivalent to [tex]\(\log \left(4^6\right)\)[/tex] is [tex]\(6 \cdot \log (4)\)[/tex].
Now, let's compare this result with the given choices:
A) [tex]\(\log (6) \cdot \log (4)\)[/tex] is not equivalent to [tex]\(6 \cdot \log (4)\)[/tex].
B) [tex]\(\log (6) + \log (4)\)[/tex] is not equivalent to [tex]\(6 \cdot \log (4)\)[/tex].
C) [tex]\(6 \cdot \log (4)\)[/tex] is exactly equivalent to our result.
D) [tex]\(\log (6) - \log (4)\)[/tex] is not equivalent to [tex]\(6 \cdot \log (4)\)[/tex].
Hence, the correct answer is:
C. [tex]\(6 \cdot \log (4)\)[/tex]
