1. **State the problem:** Solve the exponential equation $$2e^{5x} = 868$$ by taking the natural logarithm on both sides. 2. **Isolate the exponential term:** Divide both sides by 2: $$e^{5x} = \frac{868}{2}$$ $$e^{5x} = 434$$ 3. **Take the natural logarithm of both sides:** $$\ln\left(e^{5x}\right) = \ln(434)$$ 4. **Use the logarithm power rule:** $$5x \ln(e) = \ln(434)$$ Since $$\ln(e) = 1$$, this simplifies to: $$5x = \ln(434)$$ 5. **Solve for $$x$$:** $$x = \frac{\ln(434)}{5}$$ 6. **Decimal approximation:** Calculate $$\ln(434) \approx 6.072$$ Then, $$x \approx \frac{6.072}{5} = 1.2144$$ Rounded to two decimal places: $$x \approx 1.21$$ **Final answers:** - In terms of natural logarithms: $$x = \frac{\ln(434)}{5}$$ - Decimal approximation: $$x \approx 1.21$$