Answered

We can write [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], where:

[tex]A = \square[/tex]
[tex]B = \square[/tex]

Write [tex]A[/tex] and [tex]B[/tex] as integers or reduced fractions.



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]