We can write $\log _{3}\left(x \sqrt[3]{y^{14}}\right)$ into the form $A \log _{3} x+B \log _{3} y$ where $A=$ and $B=$ Write $A$ and $B$ as integers or reduced fractions. Question Help: Video Read Submit Question Jump to Answer
Solution
Step 1 ：Separate the terms inside the logarithm using the property that the logarithm of a product is the sum of the logarithms of its factors. So, \(\log _{3}\left(x \sqrt[3]{y^{14}}\right)\) can be written as \(\log _{3} x + \log _{3} \sqrt[3]{y^{14}}\).
Step 2 ：Rewrite the cube root in the second term as a power. So, \(\log _{3} \sqrt[3]{y^{14}}\) can be written as \(\log _{3} y^{14/3}\).
Step 3 ：Use the property that the logarithm of a number raised to a power is the product of the power and the logarithm of the number. So, \(\log _{3} y^{14/3}\) can be written as \(\frac{14}{3} \log _{3} y\).
Step 4 ：So, the original expression \(\log _{3}\left(x \sqrt[3]{y^{14}}\right)\) can be written as \(\log _{3} x + \frac{14}{3} \log _{3} y\).
Step 5 ：Comparing this with the form \(A \log _{3} x + B \log _{3} y\), we can see that \(A = 1\) and \(B = \frac{14}{3}\).
Step 6 ：So, the final answer is \(\boxed{A = 1}\) and \(\boxed{B = \frac{14}{3}}\).
