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]
