To write the given logarithmic expression [tex]\(\log_3\left(x^4 \sqrt[3]{y^{11}}\right)\)[/tex] in the form [tex]\(A \log_3 x + B \log_3 y\)[/tex], follow these steps:
1. Express the logarithmic argument using exponent rules:
Recall that [tex]\(\sqrt[3]{y^{11}}\)[/tex] can be written as [tex]\(y^{11/3}\)[/tex]. Therefore, the entire expression inside the logarithm is:
[tex]\[
x^4 y^{11/3}
\][/tex]
2. Apply the logarithm property:
Using the property of logarithms [tex]\(\log_b (MN) = \log_b M + \log_b N\)[/tex], we can break the logarithm of a product into the sum of logarithms:
[tex]\[
\log_3 (x^4 y^{11/3}) = \log_3 (x^4) + \log_3 (y^{11/3})
\][/tex]
3. Use the power rule of logarithms:
The power rule of logarithms states that [tex]\(\log_b (M^k) = k \log_b M\)[/tex]. Applying this to each term in the sum:
[tex]\[
\log_3 (x^4) = 4 \log_3 x
\][/tex]
[tex]\[
\log_3 (y^{11/3}) = \frac{11}{3} \log_3 y
\][/tex]
4. Combine the results:
Putting these together, we get:
[tex]\[
\log_3 (x^4 y^{11/3}) = 4 \log_3 x + \frac{11}{3} \log_3 y
\][/tex]
So, comparing this with the form [tex]\(A \log_3 x + B \log_3 y\)[/tex], we identify the coefficients [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
A = 4
\][/tex]
[tex]\[
B = \frac{11}{3}
\][/tex]
Therefore, the values of [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are:
[tex]\[
A = 4
\][/tex]
[tex]\[
B = \frac{11}{3}
\][/tex]
