To solve the problem of finding the equivalent logarithmic expression for [tex]\(\log_b(a \cdot d)\)[/tex], we should use the properties of logarithms. Specifically, we use the property:
[tex]\[
\log_b(x \cdot y) = \log_b(x) + \log_b(y)
\][/tex]
This property states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Let's apply this property step-by-step to our given expression.
1. Identify the base and the arguments:
- The base is [tex]\(b\)[/tex].
- The arguments of the logarithm are [tex]\(a\)[/tex] and [tex]\(d\)[/tex].
2. Apply the product rule of logarithms:
- According to the product rule, [tex]\(\log_b(a \cdot d)\)[/tex] can be split into two separate logarithms added together. This gives us:
[tex]\[
\log_b(a \cdot d) = \log_b(a) + \log_b(d)
\][/tex]
3. Match the result with the provided options:
- Option A: [tex]\(\log_b a + \log_b d\)[/tex]
- Option B: [tex]\(\log_b a - \log_b d\)[/tex]
- Option C: [tex]\(d \cdot \log_b a\)[/tex]
- Option D: [tex]\(\log_b a \cdot \log_b d\)[/tex]
From our derivation, we see that [tex]\(\log_b(a \cdot d)\)[/tex] is equal to [tex]\(\log_b(a) + \log_b(d)\)[/tex], which matches exactly with Option A.
Thus, the correct answer is:
A. [tex]\(\log_b a + \log_b d\)[/tex]
