1. **State the problem:** Simplify the expressions: a. $\ln e^{0.3x}$ b. $e^{2 \ln(x+3)}$ 2. **Recall the relevant formulas and rules:** - The natural logarithm and exponential functions are inverses: $\ln e^y = y$. - The power rule for logarithms: $\ln a^b = b \ln a$. - Exponentiation with logarithms: $e^{\ln a} = a$. 3. **Simplify part (a):** $$\ln e^{0.3x} = 0.3x$$ Because $\ln e^y = y$. 4. **Simplify part (b):** $$e^{2 \ln(x+3)} = e^{\ln (x+3)^2}$$ Using the power rule for logarithms: $2 \ln(x+3) = \ln (x+3)^2$. Then, since $e^{\ln a} = a$: $$e^{\ln (x+3)^2} = (x+3)^2$$ 5. **Final answers:** a. $0.3x$ b. $(x+3)^2$