Step 1 :We are given that the initial principal (P) is $2000, the final amount (A) is $4000 (since it's doubled), the annual interest rate (r) is 5.25% or 0.0525 in decimal form, and the interest is compounded monthly (n=12).
Step 2 :We need to find the time (t) it takes for the initial principal to double. We can use the formula for compound interest, which is \(A = P(1 + r/n)^{nt}\).
Step 3 :We can rearrange this formula to solve for t: \(t = \frac{\log(A/P)}{n \cdot \log(1 + r/n)}\).
Step 4 :Substituting the given values into this formula, we get \(t = \frac{\log(4000/2000)}{12 \cdot \log(1 + 0.0525/12)}\).
Step 5 :Solving this equation, we find that \(t \approx 13.23\).
Step 6 :Final Answer: It will take approximately \(\boxed{13.23}\) years for $2000 to double at an annual interest rate of 5.25% compounded monthly.