Why is the value of [tex]\log (387 \times 28)[/tex] the same as that of [tex]\log 387 + \log 28[/tex]?

Choose the correct answer below.

A. The distributive property applies to logarithms.
B. The power property of logarithms allows the two expressions to be written in an equivalent manner.
C. The logarithm of a product of two numbers is equal to the sum of the logarithms of the numbers.
D. The logarithm of a sum of two numbers is equal to the product of the logarithms of the numbers.



Answer :

To understand why [tex]\(\log (387 \times 28)\)[/tex] is equal to [tex]\(\log 387 + \log 28\)[/tex], let's take a look at the properties of logarithms, specifically the property relating to the logarithm of a product.

One of the key properties of logarithms is the Product Property, which states:

[tex]\[ \log_b (xy) = \log_b x + \log_b y \][/tex]

This property applies to any base [tex]\(b\)[/tex] and positive numbers [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. It simplifies the computation of the logarithm of a product by breaking it down into the sum of the logarithms of the individual numbers.

In this case, we have [tex]\(\log (387 \times 28)\)[/tex]. According to the Product Property of logarithms, we can write this as:

[tex]\[ \log (387 \times 28) = \log 387 + \log 28 \][/tex]

Thus, the value of [tex]\(\log (387 \times 28)\)[/tex] is the same as the sum [tex]\(\log 387 + \log 28\)[/tex]. Therefore, the correct answer is:

C. The logarithm of a product of two numbers is equal to the sum of logarithms of the numbers.