1. **State the problem:** We want to find how many years it will take for the population of an island to double if it increases by 10% each year. 2. **Formula used:** The population growth can be modeled by the formula for exponential growth: $$ P = P_0 (1 + r)^t $$ where $P$ is the population after $t$ years, $P_0$ is the original population, $r$ is the growth rate per year (as a decimal), and $t$ is the number of years. 3. **Set up the equation:** We want the population to double, so: $$ 2P_0 = P_0 (1 + 0.10)^t $$ 4. **Simplify the equation:** Divide both sides by $P_0$: $$ 2 = (1.10)^t $$ 5. **Solve for $t$ using logarithms:** Take the natural logarithm of both sides: $$ \ln(2) = \ln((1.10)^t) $$ Using the logarithm power rule: $$ \ln(2) = t \ln(1.10) $$ 6. **Isolate $t$:** $$ t = \frac{\ln(2)}{\ln(1.10)} $$ 7. **Calculate the value:** $$ t = \frac{0.6931}{0.0953} \approx 7.27 $$ 8. **Interpretation:** It will take approximately 7.27 years for the population to double at a 10% annual growth rate.