Answered

Expand your knowledge base with the help of IDNLearn.com's extensive answer archive. Discover the reliable solutions you need with help from our comprehensive and accurate Q&A platform.

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

Sagot :

GinnyAnsweridn
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.
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for trusting IDNLearn.com. We’re dedicated to providing accurate answers, so visit us again for more solutions.