1. **Problem:** Solve $\log 2x = -1$. Formula: $\log a = b \implies a = 10^b$. Step 1: Rewrite as $2x = 10^{-1}$. Step 2: Simplify $10^{-1} = \frac{1}{10}$. Step 3: Solve for $x$: $$x = \frac{\cancel{2}x}{\cancel{2}} = \frac{1}{10 \times 2} = \frac{1}{20}$$. 2. **Problem:** Solve $2\log x = -1$. Step 1: Divide both sides by 2: $$\log x = \frac{\cancel{ -1}}{\cancel{2}} = -\frac{1}{2}$$. Step 2: Rewrite as $x = 10^{-\frac{1}{2}} = \frac{1}{\sqrt{10}}$. 3. **Problem:** Solve $\log (3x + 1) = 2$. Step 1: Rewrite as $3x + 1 = 10^2 = 100$. Step 2: Solve for $x$: $$3x = 100 - 1 = 99$$. Step 3: $$x = \frac{\cancel{99}}{\cancel{3}} = 33$$. 4. **Problem:** Solve $\log x + 4 = 8$. Step 1: Isolate $\log x$: $$\log x = 8 - 4 = 4$$. Step 2: Rewrite as $x = 10^4 = 10000$. 5. **Problem:** Solve $\log 6x - 3 = -4$. Step 1: Isolate $\log 6x$: $$\log 6x = -4 + 3 = -1$$. Step 2: Rewrite as $6x = 10^{-1} = \frac{1}{10}$. Step 3: Solve for $x$: $$x = \frac{\cancel{\frac{1}{10}}}{\cancel{6}} = \frac{1}{60}$$. 6. **Problem:** Solve $\log (x - 2) = 1$. Step 1: Rewrite as $x - 2 = 10^1 = 10$. Step 2: Solve for $x$: $$x = 10 + 2 = 12$$. 7. **Problem:** Solve $\log x - \log 3 = 8$. Step 1: Use log subtraction rule: $\log \frac{x}{3} = 8$. Step 2: Rewrite as $\frac{x}{3} = 10^8$. Step 3: Solve for $x$: $$x = 3 \times 10^8 = 300000000$$. 8. **Problem:** Solve $2\log x + \log 4 = 2$. Step 1: Use log power rule: $2\log x = \log x^2$. Step 2: Combine logs: $\log x^2 + \log 4 = \log (4x^2)$. Step 3: Equation becomes $\log (4x^2) = 2$. Step 4: Rewrite as $4x^2 = 10^2 = 100$. Step 5: Solve for $x^2$: $$x^2 = \frac{\cancel{100}}{\cancel{4}} = 25$$. Step 6: Take square root: $x = \pm 5$. Step 7: Check domain: $x > 0$ for $\log x$, so $x = 5$. 9. **Problem:** Solve $\log 5 - \log 2x = 1$. Step 1: Use log subtraction rule: $\log \frac{5}{2x} = 1$. Step 2: Rewrite as $\frac{5}{2x} = 10^1 = 10$. Step 3: Solve for $x$: $$2x = \frac{5}{10} = \frac{1}{2}$$. Step 4: $$x = \frac{\cancel{\frac{1}{2}}}{\cancel{2}} = \frac{1}{4}$$. **Final answers:** 1) $\frac{1}{20}$ 2) $\frac{1}{\sqrt{10}}$ 3) $33$ 4) $10000$ 5) $\frac{1}{60}$ 6) $12$ 7) $300000000$ 8) $5$ 9) $\frac{1}{4}$