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Sam is proving the product property of logarithms.

[tex]\[
\begin{tabular}{|l|l|}
\hline
Step & Justification \\
\hline
$\log_8(MN)$ & Given \\
\hline
$=\log_3\left(3^x \cdot b^y\right)$ & Substitution \\
\hline
& \\
\hline
\end{tabular}
\][/tex]

Which expression and justification completes the third step of her proof?

A. [tex]$\log_b\left(3^n\right)$[/tex]; power rule of exponents
B. [tex]$\log_8\left(b^{x-y}\right)$[/tex]; subtraction property of exponents
C. [tex]$\log_s\left(3^{x+y}\right)$[/tex]; multiplication rule of exponents
D. [tex]$\log_8\left(b^{\frac{x}{y}}\right)$[/tex]; division property of exponents



Answer :

GinnyAnswer
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.

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