1. Solve equation (13): $2^x - 5 = 10$ Add 5 to both sides: $$2^x - 5 + 5 = 10 + 5$$ $$2^x = 15$$ Take the logarithm base 2 of both sides: $$x = \log_2 15$$ 2. Solve equation (14): $7(2^x) + 12 = 40$ Subtract 12 from both sides: $$7(2^x) + 12 - 12 = 40 - 12$$ $$7(2^x) = 28$$ Divide both sides by 7: $$\frac{7(2^x)}{\cancel{7}} = \frac{28}{\cancel{7}}$$ $$2^x = 4$$ Take logarithm base 2: $$x = \log_2 4 = 2$$ 3. Solve equation (15): $\log_6 4 + \log_6 x = 2$ Use logarithm addition rule: $$\log_6 (4x) = 2$$ Rewrite in exponential form: $$4x = 6^2 = 36$$ Divide both sides by 4: $$\frac{4x}{\cancel{4}} = \frac{36}{\cancel{4}}$$ $$x = 9$$ 4. Solve equation (16): $\log 7 + \log x = 4$ Use logarithm addition rule: $$\log (7x) = 4$$ Rewrite in exponential form (base 10): $$7x = 10^4 = 10000$$ Divide both sides by 7: $$\frac{7x}{\cancel{7}} = \frac{10000}{\cancel{7}}$$ $$x \approx 1428.57$$ 5. Solve equation (17): $4(2^{2x}) + 3 = 15$ Subtract 3 from both sides: $$4(2^{2x}) = 12$$ Divide both sides by 4: $$\frac{4(2^{2x})}{\cancel{4}} = \frac{12}{\cancel{4}}$$ $$2^{2x} = 3$$ Rewrite $2^{2x} = (2^x)^2$: $$(2^x)^2 = 3$$ Take square root: $$2^x = \sqrt{3}$$ Take logarithm base 2: $$x = \log_2 \sqrt{3} = \frac{1}{2} \log_2 3$$ 6. Solve equation (18): $\log(3x - 5) = 1$ Rewrite in exponential form (base 10): $$3x - 5 = 10^1 = 10$$ Add 5 to both sides: $$3x = 15$$ Divide both sides by 3: $$\frac{3x}{\cancel{3}} = \frac{15}{\cancel{3}}$$ $$x = 5$$ 7. Solve equation (19): $2 \log x + \log 5 = \log(14x + 3)$ Use power rule: $$\log x^2 + \log 5 = \log(14x + 3)$$ Use addition rule: $$\log (5x^2) = \log(14x + 3)$$ Since logs are equal, arguments are equal: $$5x^2 = 14x + 3$$ Bring all terms to one side: $$5x^2 - 14x - 3 = 0$$ Use quadratic formula: $$x = \frac{14 \pm \sqrt{(-14)^2 - 4 \cdot 5 \cdot (-3)}}{2 \cdot 5} = \frac{14 \pm \sqrt{196 + 60}}{10} = \frac{14 \pm \sqrt{256}}{10}$$ $$x = \frac{14 \pm 16}{10}$$ Two solutions: $$x = \frac{14 + 16}{10} = 3$$ $$x = \frac{14 - 16}{10} = -0.2$$ Check domain: $x > 0$, so $x = 3$ is valid. 8. Solve equation (20): $\log_4 (\log_3 x) = 0$ Rewrite in exponential form: $$\log_3 x = 4^0 = 1$$ Rewrite again: $$x = 3^1 = 3$$ 9. Solve equation (21): $\log_6 [\log_5 (\log_3 x)] = 0$ Rewrite outer log: $$\log_5 (\log_3 x) = 6^0 = 1$$ Rewrite inner log: $$\log_3 x = 5^1 = 5$$ Rewrite innermost: $$x = 3^5 = 243$$ Final answers: (13) $x = \log_2 15$ (14) $x = 2$ (15) $x = 9$ (16) $x \approx 1428.57$ (17) $x = \frac{1}{2} \log_2 3$ (18) $x = 5$ (19) $x = 3$ (20) $x = 3$ (21) $x = 243$