1. Problem: Calculate $\log_3 x$ for given expressions of $x$ with $a,b,c>0$ and base 3. 2. Recall logarithm rules: - $\log_b(mn) = \log_b m + \log_b n$ - $\log_b(m^k) = k \log_b m$ - $\log_b \sqrt[n]{m} = \frac{1}{n} \log_b m$ 3. Part (a): $x = \sqrt[3]{a^2} \cdot \sqrt[4]{b^3} \cdot \sqrt[3]{c}$ $$\log_3 x = \log_3 \left(a^{\frac{2}{3}} \cdot b^{\frac{3}{4}} \cdot c^{\frac{1}{3}}\right) = \log_3 a^{\frac{2}{3}} + \log_3 b^{\frac{3}{4}} + \log_3 c^{\frac{1}{3}}$$ $$= \frac{2}{3} \log_3 a + \frac{3}{4} \log_3 b + \frac{1}{3} \log_3 c$$ 4. Part (b): $x = \sqrt[5]{27 a^3 \sqrt{a b^2}} \cdot \sqrt[4]{c^3} \cdot \sqrt{a}$ First simplify inside the fifth root: $$\sqrt{a b^2} = (a b^2)^{\frac{1}{2}} = a^{\frac{1}{2}} b$$ So inside fifth root: $$27 a^3 \sqrt{a b^2} = 27 a^3 \cdot a^{\frac{1}{2}} b = 27 a^{3 + \frac{1}{2}} b = 27 a^{\frac{7}{2}} b$$ Then: $$x = \left(27 a^{\frac{7}{2}} b\right)^{\frac{1}{5}} \cdot c^{\frac{3}{4}} \cdot a^{\frac{1}{2}}$$ Apply log: $$\log_3 x = \log_3 \left(27^{\frac{1}{5}} a^{\frac{7}{10}} b^{\frac{1}{5}} c^{\frac{3}{4}} a^{\frac{1}{2}}\right)$$ $$= \log_3 27^{\frac{1}{5}} + \log_3 a^{\frac{7}{10}} + \log_3 b^{\frac{1}{5}} + \log_3 c^{\frac{3}{4}} + \log_3 a^{\frac{1}{2}}$$ $$= \frac{1}{5} \log_3 27 + \frac{7}{10} \log_3 a + \frac{1}{5} \log_3 b + \frac{3}{4} \log_3 c + \frac{1}{2} \log_3 a$$ Since $27 = 3^3$, $\log_3 27 = 3$, so: $$= \frac{1}{5} \cdot 3 + \left(\frac{7}{10} + \frac{1}{2}\right) \log_3 a + \frac{1}{5} \log_3 b + \frac{3}{4} \log_3 c$$ $$= \frac{3}{5} + \frac{7}{10} \log_3 a + \frac{5}{10} \log_3 a + \frac{1}{5} \log_3 b + \frac{3}{4} \log_3 c$$ $$= \frac{3}{5} + \frac{12}{10} \log_3 a + \frac{1}{5} \log_3 b + \frac{3}{4} \log_3 c = \frac{3}{5} + \frac{6}{5} \log_3 a + \frac{1}{5} \log_3 b + \frac{3}{4} \log_3 c$$ 5. Part (e): $x = a^3 + b^3$ Since $\log_3 (a^3 + b^3)$ cannot be simplified using log addition rules (log of sum is not sum of logs), the expression remains: $$\log_3 (a^3 + b^3)$$ Final answers: (a) $\log_3 x = \frac{2}{3} \log_3 a + \frac{3}{4} \log_3 b + \frac{1}{3} \log_3 c$ (b) $\log_3 x = \frac{3}{5} + \frac{6}{5} \log_3 a + \frac{1}{5} \log_3 b + \frac{3}{4} \log_3 c$ (e) $\log_3 x = \log_3 (a^3 + b^3)$