Sure, let's simplify the expression [tex]\(\log_c(u^3 v^4)\)[/tex] by breaking it into a sum or difference of logarithms and expressing powers as factors.
Given expression:
[tex]\[
\log_c(u^3 v^4)
\][/tex]
1. First, use the product rule of logarithms:
The product rule states that [tex]\(\log_b(A \cdot B) = \log_b(A) + \log_b(B)\)[/tex].
So, we can write:
[tex]\[
\log_c(u^3 v^4) = \log_c(u^3) + \log_c(v^4)
\][/tex]
2. Next, use the power rule of logarithms:
The power rule states that [tex]\(\log_b(A^n) = n \cdot \log_b(A)\)[/tex].
Applying this rule to each term, we get:
[tex]\[
\log_c(u^3) = 3 \cdot \log_c(u)
\][/tex]
and
[tex]\[
\log_c(v^4) = 4 \cdot \log_c(v)
\][/tex]
3. Combine the results:
Now, we can combine the two terms:
[tex]\[
\log_c(u^3 v^4) = 3 \cdot \log_c(u) + 4 \cdot \log_c(v)
\][/tex]
So, the simplified answer is:
[tex]\[
\log_c(u^3 v^4) = 3 \cdot \log_c(u) + 4 \cdot \log_c(v)
\][/tex]
