1. **State the problem:** We are given the function $$P(t) = 800(1.05)^t$$ and asked to find the value of $$t$$ when $$P(t) = 1000$$. 2. **Write the equation:** Set $$P(t) = 1000$$: $$1000 = 800(1.05)^t$$ 3. **Isolate the exponential term:** Divide both sides by 800: $$\frac{1000}{800} = \cancel{\frac{800}{800}}(1.05)^t$$ $$1.25 = (1.05)^t$$ 4. **Solve for $$t$$ using logarithms:** Take the natural logarithm (ln) of both sides: $$\ln(1.25) = \ln((1.05)^t)$$ 5. **Use logarithm power rule:** $$\ln(1.25) = t \ln(1.05)$$ 6. **Isolate $$t$$:** $$t = \frac{\ln(1.25)}{\ln(1.05)}$$ 7. **Calculate the values:** $$t = \frac{0.223143551}{0.048790164} \approx 4.57$$ 8. **Round to nearest tenth:** $$t \approx 4.6$$ **Final answer:** $$t \approx 4.6$$