To write the expression [tex]\( 7 \log_c(y-4) - 4 \log_c(y+4) \)[/tex] as a single logarithm, we can use properties of logarithms. Here are the properties that will help:
1. The power rule: [tex]\( a \log_b(x) = \log_b(x^a) \)[/tex]
2. The quotient rule: [tex]\( \log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right) \)[/tex]
Let's break it down step by step:
1. Apply the power rule to each term in the expression:
[tex]\[
7 \log_c(y-4) \implies \log_c((y-4)^7)
\][/tex]
[tex]\[
4 \log_c(y+4) \implies \log_c((y+4)^4)
\][/tex]
Now the expression becomes:
[tex]\[
\log_c((y-4)^7) - \log_c((y+4)^4)
\][/tex]
2. Apply the quotient rule to combine the two logarithms into a single logarithm:
[tex]\[
\log_c((y-4)^7) - \log_c((y+4)^4) \implies \log_c\left(\frac{(y-4)^7}{(y+4)^4}\right)
\][/tex]
Therefore, the expression as a single logarithm is:
[tex]\[
\log_c\left(\frac{(y-4)^7}{(y+4)^4}\right)
\][/tex]
