1. **State the problem:** Solve the equation $$2000 \times (1.02)^x = 2500$$ for $x$. 2. **Isolate the exponential term:** Divide both sides by 2000 to get: $$\cancel{2000} \times (1.02)^x = \frac{2500}{\cancel{2000}}$$ which simplifies to $$ (1.02)^x = \frac{2500}{2000} = 1.25 $$ 3. **Use logarithms to solve for $x$:** Taking the natural logarithm (ln) of both sides: $$ \ln\left((1.02)^x\right) = \ln(1.25) $$ 4. **Apply logarithm power rule:** $$ x \times \ln(1.02) = \ln(1.25) $$ 5. **Solve for $x$:** $$ x = \frac{\ln(1.25)}{\ln(1.02)} $$ 6. **Calculate the numerical value:** $$ x \approx \frac{0.223143551}{0.019802627} \approx 11.27 $$ **Final answer:** $$ x \approx 11.27 $$