1. Problem 8a: Solve $\log_{10} x = 2$. Formula: $\log_b a = c \implies a = b^c$. Step 1: Apply the formula: $x = 10^2$. Step 2: Calculate: $x = 100$. 2. Problem 8b: Solve $\log_x \frac{2}{3} = -\frac{1}{2}$. Step 1: Rewrite: $\frac{2}{3} = x^{-\frac{1}{2}}$. Step 2: Express as $\frac{2}{3} = \frac{1}{\sqrt{x}}$. Step 3: Invert both sides: $\sqrt{x} = \frac{3}{2}$. Step 4: Square both sides: $x = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$. 3. Problem 8c: Solve $\log_4 8 = x$. Step 1: Rewrite: $8 = 4^x$. Step 2: Express bases as powers of 2: $8 = 2^3$, $4 = 2^2$. Step 3: Substitute: $2^3 = (2^2)^x = 2^{2x}$. Step 4: Equate exponents: $3 = 2x \implies x = \frac{3}{2}$. 4. Problem 8d: Solve $\log_3 (2x - 5) = 2$. Step 1: Rewrite: $2x - 5 = 3^2 = 9$. Step 2: Solve: $2x = 14 \implies x = 7$. 5. Problem 8e: Solve $\ln(\ln x) = 1$. Step 1: Rewrite: $\ln x = e^1 = e$. Step 2: Solve: $x = e^e$. 6. Problem 8f: Solve $\log_7 (\log_4 x^2) = 0$. Step 1: Rewrite: $\log_4 x^2 = 7^0 = 1$. Step 2: Rewrite inner log: $x^2 = 4^1 = 4$. Step 3: Solve: $x = \pm 2$. 7. Problem 8g: Solve $\log_2 x + \log_2 (x + 2) = 3$. Step 1: Use log addition: $\log_2 [x(x+2)] = 3$. Step 2: Rewrite: $x(x+2) = 2^3 = 8$. Step 3: Expand: $x^2 + 2x = 8$. Step 4: Rearrange: $x^2 + 2x - 8 = 0$. Step 5: Factor: $(x+4)(x-2) = 0$. Step 6: Solutions: $x = -4$ (reject, log undefined), $x = 2$. 8. Problem 8h: Solve $\log_3 (x + 12) - \log_3 (x - 3) = \log_3 6$. Step 1: Use log subtraction: $\log_3 \frac{x+12}{x-3} = \log_3 6$. Step 2: Equate arguments: $\frac{x+12}{x-3} = 6$. Step 3: Cross multiply: $x + 12 = 6(x - 3)$. Step 4: Expand: $x + 12 = 6x - 18$. Step 5: Rearrange: $12 + 18 = 6x - x$. Step 6: $30 = 5x \implies x = 6$. 9. Problem 8i: Solve $\log_7 (x + 1) + \log_7 (x - 5) = 1$. Step 1: Use log addition: $\log_7 [(x+1)(x-5)] = 1$. Step 2: Rewrite: $(x+1)(x-5) = 7^1 = 7$. Step 3: Expand: $x^2 - 5x + x - 5 = 7$. Step 4: Simplify: $x^2 - 4x - 5 = 7$. Step 5: Rearrange: $x^2 - 4x - 12 = 0$. Step 6: Factor or use quadratic formula: $\Delta = (-4)^2 - 4(1)(-12) = 16 + 48 = 64$. Step 7: $x = \frac{4 \pm 8}{2}$. Step 8: Solutions: $x = 6$ or $x = -2$ (reject, log undefined). 10. Problem 8j: Solve $\log_2 (x + 2) + \log_2 (x - 5) = 3$. Step 1: Use log addition: $\log_2 [(x+2)(x-5)] = 3$. Step 2: Rewrite: $(x+2)(x-5) = 2^3 = 8$. Step 3: Expand: $x^2 - 5x + 2x - 10 = 8$. Step 4: Simplify: $x^2 - 3x - 10 = 8$. Step 5: Rearrange: $x^2 - 3x - 18 = 0$. Step 6: Use quadratic formula: $\Delta = (-3)^2 - 4(1)(-18) = 9 + 72 = 81$. Step 7: $x = \frac{3 \pm 9}{2}$. Step 8: Solutions: $x = 6$ or $x = -3$ (reject, log undefined). 11. Problem 8k: Solve $2 \ln x = \ln (2x + \frac{5}{2}) + \ln 2$. Step 1: Use log addition: $2 \ln x = \ln [2(2x + \frac{5}{2})]$. Step 2: Simplify inside log: $2(2x + \frac{5}{2}) = 4x + 5$. Step 3: Rewrite left side: $2 \ln x = \ln x^2$. Step 4: Equation: $\ln x^2 = \ln (4x + 5)$. Step 5: Equate arguments: $x^2 = 4x + 5$. Step 6: Rearrange: $x^2 - 4x - 5 = 0$. Step 7: Factor: $(x - 5)(x + 1) = 0$. Step 8: Solutions: $x = 5$ or $x = -1$ (reject, ln undefined). 12. Problem 8l: Solve $4 \log_x 2 - \frac{1}{2} \log_x 4 = 2 - \frac{1}{2} \log_x 8$. Step 1: Express logs in terms of $\log_x 2 = a$. Note: $\log_x 4 = \log_x 2^2 = 2a$, $\log_x 8 = \log_x 2^3 = 3a$. Step 2: Substitute: $4a - \frac{1}{2} (2a) = 2 - \frac{1}{2} (3a)$. Step 3: Simplify: $4a - a = 2 - \frac{3a}{2}$. Step 4: $3a = 2 - \frac{3a}{2}$. Step 5: Add $\frac{3a}{2}$ to both sides: $3a + \frac{3a}{2} = 2$. Step 6: $\frac{6a}{2} + \frac{3a}{2} = 2 \implies \frac{9a}{2} = 2$. Step 7: Solve for $a$: $a = \frac{4}{9}$. Step 8: Recall $a = \log_x 2 = \frac{\ln 2}{\ln x} = \frac{4}{9}$. Step 9: Solve for $\ln x$: $\ln x = \frac{9}{4} \ln 2$. Step 10: Exponentiate: $x = e^{\frac{9}{4} \ln 2} = 2^{\frac{9}{4}}$. --- 13. Problem 9a: Solve $4^x = 8$. Step 1: Express bases as powers of 2: $4 = 2^2$, $8 = 2^3$. Step 2: Rewrite: $(2^2)^x = 2^3$. Step 3: Simplify: $2^{2x} = 2^3$. Step 4: Equate exponents: $2x = 3 \implies x = \frac{3}{2}$. 14. Problem 9b: Solve $9^x = \frac{1}{243}$. Step 1: Express bases as powers of 3: $9 = 3^2$, $243 = 3^5$. Step 2: Rewrite: $(3^2)^x = 3^{-5}$. Step 3: Simplify: $3^{2x} = 3^{-5}$. Step 4: Equate exponents: $2x = -5 \implies x = -\frac{5}{2}$. 15. Problem 9c: Solve $\left(\frac{3}{2}\right)^{2x - 1} = \frac{27}{8}$. Step 1: Express right side as powers: $27 = 3^3$, $8 = 2^3$. Step 2: Rewrite: $\left(\frac{3}{2}\right)^{2x - 1} = \left(\frac{3}{2}\right)^3$. Step 3: Equate exponents: $2x - 1 = 3$. Step 4: Solve: $2x = 4 \implies x = 2$. 16. Problem 9d: Solve $3^{4x - 7} = \frac{1}{9}$. Step 1: Express $9 = 3^2$. Step 2: Rewrite: $3^{4x - 7} = 3^{-2}$. Step 3: Equate exponents: $4x - 7 = -2$. Step 4: Solve: $4x = 5 \implies x = \frac{5}{4}$. 17. Problem 9e: Solve $2^x = 5$. Step 1: Take natural log: $x \ln 2 = \ln 5$. Step 2: Solve: $x = \frac{\ln 5}{\ln 2}$. 18. Problem 9f: Solve $5^{3x - 1} = 15$. Step 1: Express $15 = 3 \times 5$. Step 2: Rewrite: $5^{3x - 1} = 3 \times 5^1$. Step 3: Divide both sides by $5$: $5^{3x - 2} = 3$. Step 4: Take natural log: $(3x - 2) \ln 5 = \ln 3$. Step 5: Solve: $3x - 2 = \frac{\ln 3}{\ln 5}$. Step 6: $3x = 2 + \frac{\ln 3}{\ln 5}$. Step 7: $x = \frac{2}{3} + \frac{1}{3} \cdot \frac{\ln 3}{\ln 5}$. 19. Problem 9g: Solve $6^x = 5 \cdot 3^{x + 1}$. Step 1: Express $6 = 2 \times 3$. Step 2: Rewrite: $(2 \times 3)^x = 5 \times 3^{x+1}$. Step 3: Expand left: $2^x \times 3^x = 5 \times 3^x \times 3^1$. Step 4: Cancel $3^x$ both sides: $2^x = 5 \times 3$. Step 5: Simplify right: $2^x = 15$. Step 6: Take natural log: $x \ln 2 = \ln 15$. Step 7: Solve: $x = \frac{\ln 15}{\ln 2}$. 20. Problem 9h: Solve $3^{2x - 1} = 5^{2 - 3x}$. Step 1: Take natural log both sides: $(2x - 1) \ln 3 = (2 - 3x) \ln 5$. Step 2: Expand: $2x \ln 3 - \ln 3 = 2 \ln 5 - 3x \ln 5$. Step 3: Group $x$ terms: $2x \ln 3 + 3x \ln 5 = 2 \ln 5 + \ln 3$. Step 4: Factor $x$: $x (2 \ln 3 + 3 \ln 5) = 2 \ln 5 + \ln 3$. Step 5: Solve: $x = \frac{2 \ln 5 + \ln 3}{2 \ln 3 + 3 \ln 5}$. 21. Problem 9i: Solve $10^{\log(2x + 7)} = 8$. Step 1: Use property: $10^{\log y} = y$. Step 2: So, $2x + 7 = 8$. Step 3: Solve: $2x = 1 \implies x = \frac{1}{2}$. 22. Problem 9j: Solve $e^{\ln(x - 1)} = 4$. Step 1: Use property: $e^{\ln y} = y$. Step 2: So, $x - 1 = 4$. Step 3: Solve: $x = 5$. 23. Problem 9k: Solve $3^{2x} + 2 = 3^{x + 1}$. Step 1: Rewrite: $3^{2x} + 2 = 3^x \times 3^1 = 3 \times 3^x$. Step 2: Let $y = 3^x$. Step 3: Equation: $y^2 + 2 = 3y$. Step 4: Rearrange: $y^2 - 3y + 2 = 0$. Step 5: Factor: $(y - 1)(y - 2) = 0$. Step 6: Solutions: $y = 1$ or $y = 2$. Step 7: Recall $y = 3^x$. Step 8: For $y=1$: $3^x = 1 \implies x=0$. Step 9: For $y=2$: $3^x = 2 \implies x = \frac{\ln 2}{\ln 3}$. 24. Problem 9l: Solve $5^{2x} - 5^{x + 1} + 4 = 0$. Step 1: Let $y = 5^x$. Step 2: Rewrite: $y^2 - 5y + 4 = 0$. Step 3: Factor: $(y - 4)(y - 1) = 0$. Step 4: Solutions: $y = 4$ or $y = 1$. Step 5: Recall $y = 5^x$. Step 6: For $y=4$: $5^x = 4 \implies x = \frac{\ln 4}{\ln 5}$. Step 7: For $y=1$: $5^x = 1 \implies x = 0$. 25. Problem 9m: Solve $4^{x+1} + 2^x = \frac{1}{2}$. Step 1: Express $4^{x+1} = (2^2)^{x+1} = 2^{2x + 2}$. Step 2: Rewrite: $2^{2x + 2} + 2^x = \frac{1}{2} = 2^{-1}$. Step 3: Let $y = 2^x$. Step 4: Equation: $2^2 y^2 + y = 2^{-1}$. Step 5: Simplify: $4 y^2 + y = \frac{1}{2}$. Step 6: Multiply both sides by 2: $8 y^2 + 2 y = 1$. Step 7: Rearrange: $8 y^2 + 2 y - 1 = 0$. Step 8: Use quadratic formula: $y = \frac{-2 \pm \sqrt{4 + 32}}{16} = \frac{-2 \pm \sqrt{36}}{16}$. Step 9: Solutions: $y = \frac{-2 + 6}{16} = \frac{4}{16} = \frac{1}{4}$ or $y = \frac{-2 - 6}{16} = -\frac{8}{16} = -\frac{1}{2}$ (reject, $y=2^x>0$). Step 10: $2^x = \frac{1}{4} = 2^{-2}$. Step 11: Solve: $x = -2$. 26. Problem 9n: Solve $\frac{e^x + e^{-x}}{2} = \frac{5}{4}$. Step 1: Recognize left side as $\cosh x$. Step 2: Multiply both sides by 2: $e^x + e^{-x} = \frac{5}{2}$. Step 3: Let $y = e^x$, then $e^{-x} = \frac{1}{y}$. Step 4: Equation: $y + \frac{1}{y} = \frac{5}{2}$. Step 5: Multiply both sides by $y$: $y^2 - \frac{5}{2} y + 1 = 0$. Step 6: Multiply entire equation by 2: $2 y^2 - 5 y + 2 = 0$. Step 7: Use quadratic formula: $y = \frac{5 \pm \sqrt{25 - 16}}{4} = \frac{5 \pm 3}{4}$. Step 8: Solutions: $y = 2$ or $y = \frac{1}{2}$. Step 9: Recall $y = e^x$. Step 10: Solve: $x = \ln 2$ or $x = \ln \frac{1}{2} = -\ln 2$. 27. Problem 9o: Solve $\frac{e^{-x}}{e^x - e^{-x}} = 2$. Step 1: Multiply both sides by denominator: $e^{-x} = 2 (e^x - e^{-x})$. Step 2: Expand right side: $e^{-x} = 2 e^x - 2 e^{-x}$. Step 3: Add $2 e^{-x}$ to both sides: $e^{-x} + 2 e^{-x} = 2 e^x$. Step 4: Simplify left side: $3 e^{-x} = 2 e^x$. Step 5: Multiply both sides by $e^x$: $3 = 2 e^{2x}$. Step 6: Solve for $e^{2x}$: $e^{2x} = \frac{3}{2}$. Step 7: Take natural log: $2x = \ln \frac{3}{2}$. Step 8: Solve: $x = \frac{1}{2} \ln \frac{3}{2}$. Final answers: 8a: $x=100$ 8b: $x=\frac{9}{4}$ 8c: $x=\frac{3}{2}$ 8d: $x=7$ 8e: $x=e^e$ 8f: $x=\pm 2$ 8g: $x=2$ 8h: $x=6$ 8i: $x=6$ 8j: $x=6$ 8k: $x=5$ 8l: $x=2^{\frac{9}{4}}$ 9a: $x=\frac{3}{2}$ 9b: $x=-\frac{5}{2}$ 9c: $x=2$ 9d: $x=\frac{5}{4}$ 9e: $x=\frac{\ln 5}{\ln 2}$ 9f: $x=\frac{2}{3} + \frac{1}{3} \frac{\ln 3}{\ln 5}$ 9g: $x=\frac{\ln 15}{\ln 2}$ 9h: $x=\frac{2 \ln 5 + \ln 3}{2 \ln 3 + 3 \ln 5}$ 9i: $x=\frac{1}{2}$ 9j: $x=5$ 9k: $x=0$ or $x=\frac{\ln 2}{\ln 3}$ 9l: $x=\frac{\ln 4}{\ln 5}$ or $x=0$ 9m: $x=-2$ 9n: $x=\ln 2$ or $x=-\ln 2$ 9o: $x=\frac{1}{2} \ln \frac{3}{2}$