1. **State the problem:** We are given data points for the value of an investment $x$ and the number of years $y$ it took to reach that value. We want to find the logarithmic regression equation of the form $$y = a + b \ln x$$ and then use it to find how many complete years it will take for the investment to double from $1000$ to $2000$. 2. **Set up the regression equation:** The model is $$y = a + b \ln x$$ where $y$ is the number of years and $x$ is the investment value. 3. **Calculate $\ln x$ for each $x$ value:** $$\ln 1082 \approx 6.985$$ $$\ln 1170 \approx 7.064$$ $$\ln 1265 \approx 7.143$$ $$\ln 1369 \approx 7.222$$ $$\ln 1480 \approx 7.301$$ 4. **Use the points $(\ln x, y)$ to find $a$ and $b$ by linear regression:** We have points: $$(6.985, 2), (7.064, 4), (7.143, 6), (7.222, 8), (7.301, 10)$$ Calculate means: $$\bar{X} = \frac{6.985 + 7.064 + 7.143 + 7.222 + 7.301}{5} = 7.143$$ $$\bar{Y} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6$$ Calculate slope $b$: $$b = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2}$$ Calculate numerator: $$(6.985 - 7.143)(2 - 6) + (7.064 - 7.143)(4 - 6) + (7.143 - 7.143)(6 - 6) + (7.222 - 7.143)(8 - 6) + (7.301 - 7.143)(10 - 6)$$ $$= (-0.158)(-4) + (-0.079)(-2) + 0 + 0.079(2) + 0.158(4)$$ $$= 0.632 + 0.158 + 0 + 0.158 + 0.632 = 1.58$$ Calculate denominator: $$(6.985 - 7.143)^2 + (7.064 - 7.143)^2 + (7.143 - 7.143)^2 + (7.222 - 7.143)^2 + (7.301 - 7.143)^2$$ $$= 0.025 + 0.006 + 0 + 0.006 + 0.025 = 0.062$$ So, $$b = \frac{1.58}{0.062} = 25.48$$ 5. **Calculate intercept $a$:** $$a = \bar{Y} - b \bar{X} = 6 - 25.48 \times 7.143 = 6 - 182.0 = -176.0$$ 6. **Write the regression equation:** $$y = -176.0 + 25.48 \ln x$$ 7. **Find the number of years to double the investment to $2000$:** Substitute $x=2000$: $$y = -176.0 + 25.48 \ln 2000$$ Calculate $\ln 2000$: $$\ln 2000 \approx 7.601$$ So, $$y = -176.0 + 25.48 \times 7.601 = -176.0 + 193.6 = 17.6$$ 8. **Interpretation:** It will take approximately 17 complete years for the investment to double in value. **Final answer:** $$\boxed{y = -176.0 + 25.48 \ln x}$$ $$\text{Years to double investment} = 17$$