1. Problem: Expand the logarithmic expressions. 2. Recall the logarithm rules: - $\log_b(AB) = \log_b A + \log_b B$ - $\log_b\left(\frac{A}{B}\right) = \log_b A - \log_b B$ - $\log_b(A^k) = k \log_b A$ 3. Expand each expression: (7) $\log_5 \left(\frac{5x^6}{y}\right) = \log_5 5 + \log_5 x^6 - \log_5 y$ Using $\log_5 5 = 1$ and power rule: $$= 1 + 6 \log_5 x - \log_5 y$$ (8) $\log_2 \left(7 \sqrt{a}\right) = \log_2 7 + \log_2 a^{1/2}$ Using power rule: $$= \log_2 7 + \frac{1}{2} \log_2 a$$ (9) $\log_8 \left(\frac{8}{xy^2}\right) = \log_8 8 - \log_8 x - \log_8 y^2$ Using $\log_8 8 = 1$ and power rule: $$= 1 - \log_8 x - 2 \log_8 y$$ 4. Problem: Solve for $x$ and round to hundredths. (10) $3^x = 15$ Take $\log$ base 3: $$x = \log_3 15 = \frac{\log 15}{\log 3}$$ Calculate: $$x \approx \frac{1.1761}{0.4771} \approx 2.47$$ (11) $\log_3 x = -4$ Rewrite as exponential: $$x = 3^{-4} = \frac{1}{3^4} = \frac{1}{81} \approx 0.01$$ (12) $\log_x 12 = 4$ Rewrite as exponential: $$12 = x^4$$ Solve for $x$: $$x = \sqrt[4]{12} = 12^{1/4}$$ Calculate: $$x \approx 1.86$$