Answer :
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]
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]