Why is the value of [tex]\log (350 \times 21)[/tex] the same as that of [tex]\log 350 + \log 21[/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 the value of [tex]\(\log (350 \times 21)\)[/tex] is the same as that of [tex]\(\log 350 + \log 21\)[/tex], let's review some fundamental properties of logarithms. One of the key properties is the Product Property of Logarithms.

The Product Property of Logarithms states that:
[tex]\[ \log_b (xy) = \log_b x + \log_b y \][/tex]
This property tells us that the logarithm of a product [tex]\(xy\)[/tex] is equal to the sum of the logarithms of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].

Now let's apply this to our specific question:

Given the expression [tex]\(\log (350 \times 21)\)[/tex], using the Product Property of Logarithms, we can split it into:
[tex]\[ \log (350 \times 21) = \log 350 + \log 21 \][/tex]

From this property, we see that the expression [tex]\(\log (350 \times 21)\)[/tex] is equivalent to [tex]\(\log 350 + \log 21\)[/tex].

Thus, the correct answer is:

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