To solve the problem [tex]\(\log _b(a^d)\)[/tex], let's use the properties of logarithms. Recall that for any positive real numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex], and any real number [tex]\( d \)[/tex],
[tex]\[ \log_b(a^d) \][/tex]
we can apply the logarithm power rule. The logarithm power rule states that:
[tex]\[ \log_b(a^d) = d \cdot \log_b(a) \][/tex]
So, if we have the expression [tex]\(\log _b(a^d)\)[/tex], it simplifies as follows:
1. Start with the given logarithmic expression: [tex]\(\log_b(a^d)\)[/tex].
2. Use the power rule of logarithms: This rule allows us to move the exponent [tex]\(d\)[/tex] from the argument of the logarithm to a coefficient in front. Therefore:
[tex]\[ \log_b(a^d) = d \cdot \log_b(a) \][/tex]
Hence, based on this simplification, [tex]\(\log _b(a^d)\)[/tex] is equal to [tex]\(d \cdot \log _b a\)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{A. \ d \cdot \log _b a} \][/tex]
