Expand each as much as possible. (3 Points Each) \#6: \( \log _{3}(3 x) \) \#7: \( \log _{2}(x(x+1)) \) \#8: \( \log _{\left(x^{\wedge} 5^{\star} y^{\wedge} 6 / z^{\wedge} 4\right)} \)

To expand each logarithmic expression, we’ll apply the properties of logarithms, including the product rule, the quotient rule, and the power rule. For #6: We have \( \log _{3}(3 x) \). Using the product rule of logarithms, this can be expanded as: \[ \log_{3}(3) + \log_{3}(x). \] Since \( \log_{3}(3) = 1 \), the expression simplifies to: \[ 1 + \log_{3}(x). \] For #7: Next, we consider \( \log _{2}(x(x+1)) \). Again, using the product rule, we can expand this as: \[ \log_{2}(x) + \log_{2}(x+1). \] There are no further simplifications available here, so the expression remains: \[ \log_{2}(x) + \log_{2}(x+1). \] For #8: Here we start with \( \log _{\left(x^{\wedge} 5^{\star} y^{\wedge} 6 / z^{\wedge} 4\right)} \). To expand, we’ll apply the quotient rule first, which gives us: \[ \log_{\left(x^{5} y^{6}\right)} - \log_{\left(z^{4}\right)}. \] Using the quotient rule again on the first term: \[ \log_{x^{5}}(y^{6}) = \frac{\log_{y}(y^{6})}{\log_{y}(x^{5})} = \frac{6 \log_{y}(y)}{5 \log_{y}(x)} = \frac{6}{5}, \] So the first part becomes \( 5\log_{x}(x) + 6\log_{x}(y) \). Since \( \log_{x}(x) = 1 \), we can write: \[ 5 + \frac{6 \log_{x}(y)}{5} - \frac{4 \log_{x}(z)}{4} = \frac{6 \log_{x}(y)}{5} - 6. \] Thus, the complete expansion goes as follows: \[ \frac{6\log_{x}(y)}{5} - 4. \] So, in summary, the expansions are: 1. \( 1 + \log_{3}(x) \) 2. \( \log_{2}(x) + \log_{2}(x+1) \) 3. \( 5 + \frac{6\log_{x}(y)}{5} - 4 \)