Which of the following is equivalent to the expression below?

[tex]\[
\log _8 64+\log _8 8
\][/tex]

A. [tex]\(\log _8 8\)[/tex]
B. [tex]\(\log 512\)[/tex]
C. [tex]\(\log _8 72\)[/tex]
D. [tex]\(\log _8 512\)[/tex]



Answer :

To determine which option is equivalent to the given expression [tex]\(\log_8 64 + \log_8 8\)[/tex], let's proceed with the following steps:

1. Utilize the properties of logarithms: One of the key properties is that [tex]\(\log_b(a) + \log_b(c) = \log_b(ac)\)[/tex]. This property allows us to combine the logarithms when they have the same base.

2. Apply the property to our expression:
[tex]\[ \log_8 64 + \log_8 8 = \log_8 (64 \times 8) \][/tex]

3. Simplify the expression inside the logarithm:
[tex]\[ 64 \times 8 \][/tex]
We know that:
[tex]\[ 64 = 8^2 \][/tex]
Hence:
[tex]\[ 64 \times 8 = 8^2 \times 8 = 8^3 \][/tex]

4. Rewrite the combined logarithm:
[tex]\[ \log_8 (64 \times 8) = \log_8 (8^3) \][/tex]

5. Simplify the logarithm:
[tex]\[ \log_8 (8^3) \][/tex]
Since [tex]\(\log_b(b^k) = k\)[/tex], this logarithm simplifies to:
[tex]\[ \log_8 (8^3) = 3 \][/tex]

6. Determine which answer option this corresponds to:
[tex]\[ \log_8 (8^3) = \log_8 512 \][/tex]

So, the expression [tex]\(\log_8 64 + \log_8 8\)[/tex] is equivalent to [tex]\(\log_8 512\)[/tex], which corresponds to option D:

D. [tex]\(\log_8 512\)[/tex]