1. **Problem statement:** Solve the logarithmic equations: 3) $$\log_3^2 x + \log_3 x^2 = 8$$ 4) $$\lg^3 x^2 = 8 \lg x$$ 2. **Recall important formulas and rules:** - $$\log_a b^c = c \log_a b$$ - $$\log_a^n x$$ means $$(\log_a x)^n$$ (logarithm raised to power $n$) - Let $$y = \log_3 x$$ for equation 3 and $$t = \lg x$$ for equation 4 to simplify. 3. **Solve equation 3:** Given: $$\log_3^2 x + \log_3 x^2 = 8$$ Rewrite $$\log_3 x^2 = 2 \log_3 x$$: $$y^2 + 2y = 8$$ Bring all terms to one side: $$y^2 + 2y - 8 = 0$$ Use quadratic formula: $$y = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-8)}}{2} = \frac{-2 \pm \sqrt{4 + 32}}{2} = \frac{-2 \pm \sqrt{36}}{2}$$ $$y = \frac{-2 \pm 6}{2}$$ Two solutions: $$y_1 = \frac{-2 + 6}{2} = 2$$ $$y_2 = \frac{-2 - 6}{2} = -4$$ Recall $$y = \log_3 x$$, so: $$x_1 = 3^2 = 9$$ $$x_2 = 3^{-4} = \frac{1}{81}$$ 4. **Solve equation 4:** Given: $$\lg^3 x^2 = 8 \lg x$$ Rewrite $$\lg x^2 = 2 \lg x$$: $$(2 \lg x)^3 = 8 \lg x$$ Simplify left side: $$8 (\lg x)^3 = 8 \lg x$$ Divide both sides by 8: $$(\lg x)^3 = \lg x$$ Rewrite as: $$(\lg x)^3 - \lg x = 0$$ Factor out $$\lg x$$: $$\lg x ((\lg x)^2 - 1) = 0$$ Set each factor to zero: 1) $$\lg x = 0 \Rightarrow x = 10^0 = 1$$ 2) $$(\lg x)^2 - 1 = 0 \Rightarrow (\lg x)^2 = 1 \Rightarrow \lg x = \pm 1$$ For $$\lg x = 1$$: $$x = 10^1 = 10$$ For $$\lg x = -1$$: $$x = 10^{-1} = 0.1$$ 5. **Final answers:** - For equation 3: $$x = 9$$ or $$x = \frac{1}{81}$$ - For equation 4: $$x = 1$$, $$x = 10$$, or $$x = 0.1$$