Step 1 :We are given that the principal amount (P) is $5,000, the final amount (A) is $9,000, the annual interest rate (r) is 7% or 0.07 in decimal form, and the interest is compounded monthly which means it is compounded 12 times a year (n=12). We need to find the time (t) in years it will take for the principal amount to grow to the final amount.
Step 2 :We can use the formula for compound interest which is \(A = P(1 + \frac{r}{n})^{nt}\).
Step 3 :We rearrange the formula to solve for t, which gives us \(t = \frac{\log(\frac{A}{P})}{n \cdot \log(1 + \frac{r}{n})}\).
Step 4 :Substituting the given values into the formula, we get \(t = \frac{\log(\frac{9000}{5000})}{12 \cdot \log(1 + \frac{0.07}{12})}\).
Step 5 :Solving the above expression, we find that t is approximately 8.4.
Step 6 :So, it will take approximately \(\boxed{8.4}\) years for $5,000 to grow to $9,000 if it is invested at 7% compounded monthly.