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Write the following expression as a sum and/or difference of logarithms. Express powers as factors.

[tex]\[ \log_c\left(u^3 v^4\right) \quad u \ \textgreater \ 0, v \ \textgreater \ 0 \][/tex]

[tex]\[ \log_c\left(u^3 v^4\right) = \square \][/tex]

(Simplify your answer.)

Sagot :

GinnyAnswer
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]
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