To rewrite the logarithmic expression [tex]\(\log_c(x^3 y^3 z)\)[/tex] as sums or differences of logarithms, let's use the properties of logarithms step-by-step.
1. Product Rule of Logarithms: The logarithm of a product is equal to the sum of the logarithms of the factors.
[tex]\[
\log_c(a \cdot b) = \log_c(a) + \log_c(b)
\][/tex]
Applying this rule to our expression:
[tex]\[
\log_c(x^3 y^3 z) = \log_c(x^3) + \log_c(y^3) + \log_c(z)
\][/tex]
2. Power Rule of Logarithms: The logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the base number.
[tex]\[
\log_c(a^b) = b \cdot \log_c(a)
\][/tex]
Applying this rule to each term in our expression:
[tex]\[
\log_c(x^3) = 3 \cdot \log_c(x) \\
\log_c(y^3) = 3 \cdot \log_c(y)
\][/tex]
3. Combining Results: Substitute these simplified terms back into the expression from step 1:
[tex]\[
\log_c(x^3) + \log_c(y^3) + \log_c(z) = 3 \cdot \log_c(x) + 3 \cdot \log_c(y) + \log_c(z)
\][/tex]
Therefore, the expression [tex]\(\log_c(x^3 y^3 z)\)[/tex] rewritten as sums or differences of logarithms is:
[tex]\[
\boxed{3 \cdot \log_c(x) + 3 \cdot \log_c(y) + \log_c(z)}
\][/tex]
