1. **Problem Statement:** Solve for $k$ or $t$ in each given exponential equation. 2. **Key Formula:** For equations of the form $e^{ax} = b$, take the natural logarithm on both sides to get $ax = \ln b$, then solve for $x$. 3. **Exercise 55:** a. $e^{2k} = 4$ Take natural log: $2k = \ln 4$ Solve: $k = \frac{\ln 4}{2}$ b. $100^{10k} = 200$ Rewrite base: $100 = 10^2$, so $100^{10k} = (10^2)^{10k} = 10^{20k}$ Take log base 10: $20k = \log_{10} 200$ Solve: $k = \frac{\log_{10} 200}{20}$ c. $e^{k/1000} = a$ Take natural log: $\frac{k}{1000} = \ln a$ Solve: $k = 1000 \ln a$ 4. **Exercise 56:** a. $e^{5k} = \frac{1}{4}$ Take natural log: $5k = \ln \frac{1}{4} = -\ln 4$ Solve: $k = -\frac{\ln 4}{5}$ b. $80^{ek} = 1$ Since any positive number to the zero power is 1, set exponent zero: $ek \ln 80 = 0 \Rightarrow k = 0$ c. $e^{(\ln 0.8)k} = 0.8$ Take natural log: $(\ln 0.8)k = \ln 0.8$ Solve: $k = \frac{\ln 0.8}{\ln 0.8} = 1$ 5. **Exercise 57:** a. $e^{-0.3t} = 27$ Take natural log: $-0.3t = \ln 27$ Solve: $t = -\frac{\ln 27}{0.3}$ b. $e^{kt} = \frac{1}{2}$ Take natural log: $kt = \ln \frac{1}{2} = -\ln 2$ Solve: $t = -\frac{\ln 2}{k}$ c. $e^{(\ln 0.2)t} = 0.4$ Take natural log: $(\ln 0.2)t = \ln 0.4$ Solve: $t = \frac{\ln 0.4}{\ln 0.2}$ 6. **Exercise 58:** a. $e^{-0.01t} = 1000$ Take natural log: $-0.01t = \ln 1000$ Solve: $t = -\frac{\ln 1000}{0.01}$ b. $e^{lt} = \frac{1}{10}$ Take natural log: $lt = \ln \frac{1}{10} = -\ln 10$ Solve: $t = -\frac{\ln 10}{l}$ c. $e^{(\ln 2)t} = \frac{1}{2}$ Take natural log: $(\ln 2)t = \ln \frac{1}{2} = -\ln 2$ Solve: $t = -1$ 7. **Exercise 59:** $e^{\sqrt{t}} = x^{2}$ Take natural log: $\sqrt{t} = \ln x^{2} = 2 \ln x$ Square both sides: $t = (2 \ln x)^{2} = 4 (\ln x)^{2}$ 8. **Exercise 60:** $e^{x^{2}} e^{2x+1} = e^{t}$ Use exponent addition: $e^{x^{2} + 2x + 1} = e^{t}$ So, $t = x^{2} + 2x + 1 = (x+1)^{2}$ **Final answers:** 55a. $k = \frac{\ln 4}{2}$ 55b. $k = \frac{\log_{10} 200}{20}$ 55c. $k = 1000 \ln a$ 56a. $k = -\frac{\ln 4}{5}$ 56b. $k = 0$ 56c. $k = 1$ 57a. $t = -\frac{\ln 27}{0.3}$ 57b. $t = -\frac{\ln 2}{k}$ 57c. $t = \frac{\ln 0.4}{\ln 0.2}$ 58a. $t = -\frac{\ln 1000}{0.01}$ 58b. $t = -\frac{\ln 10}{l}$ 58c. $t = -1$ 59. $t = 4 (\ln x)^{2}$ 60. $t = (x+1)^{2}$