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

Alright, let’s dive into these logarithm expressions and expand them! ### For \( \log _{3}(3 x) \): To expand \( \log _{3}(3 x) \), we can apply the properties of logarithms. We know that \( \log_b(mn) = \log_b(m) + \log_b(n) \). So, we break it down: \[ \log _{3}(3 x) = \log _{3}(3) + \log _{3}(x) \] Since \( \log _{3}(3) = 1 \) (the base and the number are the same), we get: \[ \log _{3}(3 x) = 1 + \log _{3}(x) \] ### For \( \log _{2}(x(x+1)) \): Using the same property \( \log_b(mn) = \log_b(m) + \log_b(n) \), we can expand this logarithm: \[ \log_{2}(x(x+1)) = \log_{2}(x) + \log_{2}(x+1) \] This uses the fact that the logarithm of a product is the sum of the logarithms. There is no further expansion possible since \( \log_{2}(x) \) and \( \log_{2}(x+1) \) are already in their simplest forms. ### For \( \log\left(\frac{x^{5} y^{6}}{z^{4}}\right) \): Here we will use both the property of logarithms for division \( \log_b(m/n) = \log_b(m) - \log_b(n) \) and the property for powers \( \log_b(m^n) = n \cdot \log_b(m) \). First, we handle the division: \[ \log\left(\frac{x^{5} y^{6}}{z^{4}}\right) = \log(x^5 y^6) - \log(z^4) \] Now, we expand \( \log(x^5 y^6) \) using the product property: \[ \log(x^5 y^6) = \log(x^5) + \log(y^6) = 5 \log(x) + 6 \log(y) \] Putting this all together, we get: \[ \log\left(\frac{x^{5} y^{6}}{z^{4}}\right) = (5 \log(x) + 6 \log(y)) - 4 \log(z) \] So the final expanded result is: \[ \log\left(\frac{x^{5} y^{6}}{z^{4}}\right) = 5 \log(x) + 6 \log(y) - 4 \log(z) \] And there you have it! All three logarithm expressions expanded as much as possible!