When interest is compounded (n) times a year, the cumulated amount (A) after (t)years is given by the formula a=P(1+r/n)^nt

Where P is the initial principal and r is the annual rate of interest. Approximately how long will it take 2.000$ to of double at an annual interest rate of $5.25 compounded monthly

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.