To complete the third step of Sam's proof for the product property of logarithms, we need to correctly apply the logarithm properties to the expression [tex]\( \log_8 (3^x \cdot b^y) \)[/tex]. Specifically, we should recognize that the logarithm of a product can be expressed as the sum of the logarithms of the individual factors, according to the product property of logarithms:
[tex]\[
\log_b (MN) = \log_b (M) + \log_b (N)
\][/tex]
Given the expression in the second step:
[tex]\[
\log_8 (3^x \cdot b^y)
\][/tex]
we apply the product property to get:
[tex]\[
\log_8 (3^x \cdot b^y) = \log_8 (3^x) + \log_8 (b^y)
\][/tex]
Each term can then be simplified using the power rule of logarithms, which states [tex]\( \log_b (a^c) = c \log_b (a) \)[/tex]:
[tex]\[
\log_8 (3^x) + \log_8 (b^y) = x \log_8 (3) + y \log_8 (b)
\][/tex]
Thus, the correct expression and its justification for the third step in Sam's proof are:
[tex]\[
\log_8 (3^x \cdot b^y) = x \log_8 (3) + y \log_8 (b)
\][/tex]
But since we are supposed to choose from the given options and our answer should be based on the provided solution:
The closest matching option is:
\!
So it completes the step logically given the choices available.
