1. **State the problem:** We want to find the time $t$ needed for an investment of 700000 to grow to 781900 at an interest rate of 1.3% per period. 2. **Formula used:** For compound interest, the formula is $$A = P(1 + r)^t$$ where: - $A$ is the amount after time $t$ - $P$ is the principal (initial investment) - $r$ is the interest rate per period (as a decimal) - $t$ is the time (number of periods) 3. **Plug in the known values:** $$781900 = 700000(1 + 0.013)^t$$ 4. **Isolate the exponential term:** $$\frac{781900}{700000} = (1.013)^t$$ 5. **Simplify the fraction:** $$\frac{\cancel{781900}}{\cancel{700000}} = 1.117$$ (approximate) 6. **Take the natural logarithm of both sides:** $$\ln(1.117) = \ln((1.013)^t)$$ 7. **Use logarithm power rule:** $$\ln(1.117) = t \ln(1.013)$$ 8. **Solve for $t$:** $$t = \frac{\ln(1.117)}{\ln(1.013)}$$ 9. **Calculate the values:** $$t = \frac{0.1107}{0.0129} \approx 8.58$$ **Final answer:** The time needed is approximately **8.58 periods**.