To expand [tex]\(\log \left(\frac{a^2}{b^{-3} c^4}\right)\)[/tex] as a sum, difference, or product of logarithms, we'll use the properties of logarithms which are:
1. [tex]\(\log\left(\frac{M}{N}\right) = \log(M) - \log(N)\)[/tex]
2. [tex]\(\log(M \cdot N) = \log(M) + \log(N)\)[/tex]
3. [tex]\(\log(M^k) = k \log(M)\)[/tex]
Let's break down the expression step-by-step:
1. First, recognize the structure of the expression:
[tex]\[
\log \left( \frac{a^2}{b^{-3} c^4} \right)
\][/tex]
2. Apply the quotient rule for logarithms:
[tex]\[
\log \left( \frac{a^2}{b^{-3} c^4} \right) = \log(a^2) - \log(b^{-3} c^4)
\][/tex]
3. Apply the product rule for logarithms in the second term:
[tex]\[
\log(a^2) - \log(b^{-3} c^4) = \log(a^2) - (\log(b^{-3}) + \log(c^4))
\][/tex]
4. Apply the power rule for logarithms for each term:
[tex]\[
\log(a^2) - (\log(b^{-3}) + \log(c^4)) = 2 \log(a) - (-3 \log(b) + 4 \log(c))
\][/tex]
5. Simplify the expression:
[tex]\[
2 \log(a) - (-3 \log(b) + 4 \log(c)) = 2 \log(a) - (-3 \log(b)) - 4 \log(c)
\][/tex]
[tex]\[
2 \log(a) + 3 \log(b) - 4 \log(c)
\][/tex]
Therefore, the expanded form of [tex]\(\log \left(\frac{a^2}{b^{-3} c^4}\right)\)[/tex] is:
[tex]\[
\log(a^2) + 3\log(b) - \log(c^4)
\][/tex]
or more compactly:
[tex]\[
2 \log(a) + 3 \log(b) - 4 \log(c)
\][/tex]
So, the correct and final expanded form is:
[tex]\[
2 \log(a) + 3 \log(b) - 4 \log(c)
\][/tex]
