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For any positive numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(d\)[/tex], with [tex]\(b \neq 1\)[/tex]:

[tex]\(\log_b(a \cdot d) = \)[/tex]

A. [tex]\(\log_b a + \log_b d\)[/tex]
B. [tex]\(\log_b a - \log_b d\)[/tex]
C. [tex]\(d \cdot \log_b a\)[/tex]
D. [tex]\(\log_b a \cdot \log_b d\)[/tex]



Answer :

GinnyAnswer
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]

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