Which of these expressions is equivalent to [tex]\log (6 \cdot 7)[/tex]?

A. [tex]\log (6) + \log (7)[/tex]
B. [tex]\log (6) \cdot \log (7)[/tex]
C. [tex]6 \cdot \log (7)[/tex]
D. [tex]\log (6) - \log (7)[/tex]



Answer :

To determine which expression is equivalent to [tex]\(\log (6 \cdot 7)\)[/tex], we will use the properties of logarithms.

One of the key properties of logarithms is the product rule, which states:
[tex]\[ \log_b (a \cdot c) = \log_b (a) + \log_b (c) \][/tex]

Here, we need to apply this rule to the logarithm with the given expression [tex]\(\log (6 \cdot 7)\)[/tex]. Let's break this down step by step:

1. Given expression: [tex]\(\log (6 \cdot 7)\)[/tex]

2. Apply the product rule of logarithms:
[tex]\[ \log (6 \cdot 7) = \log (6) + \log (7) \][/tex]

Thus, by applying the product rule of logarithms, we find that [tex]\(\log (6 \cdot 7)\)[/tex] is equivalent to [tex]\(\log (6) + \log (7)\)[/tex].

Let's now review the given options and identify which one matches our derived expression:

A. [tex]\(\log (6) + \log (7)\)[/tex]

B. [tex]\(\log (6) \cdot \log (7)\)[/tex]

C. [tex]\(6 \cdot \log (7)\)[/tex]

D. [tex]\(\log (6) - \log (7)\)[/tex]

As we can see, option A is:
[tex]\[ \log (6) + \log (7) \][/tex]

This matches our derived expression.

Therefore, the correct answer is:
[tex]\[ \boxed{\text{A}} \][/tex]