To transform the expression [tex]\(\log_3\left(x^{10} \sqrt[3]{y^{10}}\right)\)[/tex] into the form [tex]\(A \log_3 x + B \log_3 y\)[/tex], let's break down the process step-by-step:
1. Given Expression: [tex]\(\log_3\left(x^{10} \sqrt[3]{y^{10}}\right)\)[/tex].
2. Simplify the component involving [tex]\(y\)[/tex]:
[tex]\[
\sqrt[3]{y^{10}} = (y^{10})^{1/3} = y^{10/3}
\][/tex]
3. Next, combine the components inside the logarithm:
[tex]\[
\log_3\left(x^{10} y^{10/3}\right)
\][/tex]
4. Use the logarithm property [tex]\( \log_b(m \cdot n) = \log_b(m) + \log_b(n) \)[/tex]:
[tex]\[
\log_3(x^{10} y^{10/3}) = \log_3(x^{10}) + \log_3(y^{10/3})
\][/tex]
5. Apply the logarithm power rule [tex]\( \log_b(m^k) = k \log_b(m) \)[/tex]:
[tex]\[
\log_3(x^{10}) = 10 \log_3(x)
\][/tex]
[tex]\[
\log_3(y^{10/3}) = \frac{10}{3} \log_3(y)
\][/tex]
6. Combine the results:
[tex]\[
\log_3\left(x^{10} \sqrt[3]{y^{10}}\right) = 10 \log_3(x) + \frac{10}{3} \log_3(y)
\][/tex]
By comparing this form with [tex]\(A \log_3 x + B \log_3 y\)[/tex], we can identify:
[tex]\[
A = 10
\][/tex]
[tex]\[
B = \frac{10}{3}
\][/tex]
Thus, the constants [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are:
[tex]\[
A = 10
\][/tex]
[tex]\[
B = \frac{10}{3}
\][/tex]
