Questions: Use the Laws of Logarithms to expand the expression. log3(AB^7)

Solution Steps

To expand the logarithmic expression \(\log_{3}(A B^{7})\), we can use the Laws of Logarithms. Specifically, we will use the product rule, which states that \(\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)\), and the power rule, which states that \(\log_{b}(x^n) = n \cdot \log_{b}(x)\).

1. Apply the product rule to separate the logarithm of a product into the sum of logarithms.
2. Apply the power rule to the second term to bring the exponent down as a coefficient.

Using the product rule of logarithms, we can expand the expression \( \log_{3}(A B^{7}) \) as follows: \[ \log_{3}(A B^{7}) = \log_{3}(A) + \log_{3}(B^{7}) \]

Next, we apply the power rule to the second term \( \log_{3}(B^{7}) \): \[ \log_{3}(B^{7}) = 7 \cdot \log_{3}(B) \]

Combining the results from Steps 1 and 2, we have: \[ \log_{3}(A B^{7}) = \log_{3}(A) + 7 \cdot \log_{3}(B) \]

Final Answer