1. **State the problem:** Solve the equation $$(e^x - 2)(e^x - 3) = 0$$ for $x$. 2. **Formula and rule:** For a product of two factors to be zero, at least one of the factors must be zero. So, we set each factor equal to zero: $$e^x - 2 = 0 \quad \text{or} \quad e^x - 3 = 0$$ 3. **Solve each equation:** - For $$e^x - 2 = 0$$: $$e^x = 2$$ Taking the natural logarithm on both sides: $$\ln(e^x) = \ln(2)$$ Using the property $$\ln(e^x) = x$$: $$x = \ln(2)$$ - For $$e^x - 3 = 0$$: $$e^x = 3$$ Taking the natural logarithm on both sides: $$\ln(e^x) = \ln(3)$$ $$x = \ln(3)$$ 4. **Final answer:** $$x = \ln(2) \quad \text{or} \quad x = \ln(3)$$ These are the two solutions to the equation.