To rewrite the given logarithmic expression [tex]\(\log_c(x^8 y^4 z)\)[/tex] as sums or differences of logarithms, we can apply the product and power properties of logarithms. Here is the detailed step-by-step solution:
1. Start with the original expression:
[tex]\[
\log_c(x^8 y^4 z)
\][/tex]
2. Apply the logarithm of a product property:
The property states that:
[tex]\[
\log_b(mn) = \log_b(m) + \log_b(n)
\][/tex]
Using this property, we can split the logarithm of the product into the sum of the logarithms:
[tex]\[
\log_c(x^8 y^4 z) = \log_c(x^8) + \log_c(y^4) + \log_c(z)
\][/tex]
3. Apply the power rule of logarithms:
The power rule states that:
[tex]\[
\log_b(m^n) = n \cdot \log_b(m)
\][/tex]
Apply this rule to each term:
[tex]\[
\log_c(x^8) = 8 \cdot \log_c(x)
\][/tex]
[tex]\[
\log_c(y^4) = 4 \cdot \log_c(y)
\][/tex]
[tex]\[
\log_c(z)
\][/tex]
4. Combine all the terms:
Summing up all the terms obtained from the power rule:
[tex]\[
\log_c(x^8 y^4 z) = 8 \cdot \log_c(x) + 4 \cdot \log_c(y) + \log_c(z)
\][/tex]
So, the expression [tex]\(\log_c(x^8 y^4 z)\)[/tex] rewritten as sums of logarithms is:
[tex]\[
\log_c(x^8 y^4 z) = 8 \cdot \log_c(x) + 4 \cdot \log_c(y) + \log_c(z)
\][/tex]
