Using either logarithms or a graphing calculator, find the time required for the initial amount to be at least equal to the final amount. $\$ 8000$, deposited at $7.4 \%$ compounded monthly, to reach at least $\$ 11,000$ The time required is year(s) and months.

Step 1 ：We are given that the initial amount (P) is $8000, the final amount (A) is $11000, the annual interest rate (r) is 7.4% or 0.074 in decimal form, and the interest is compounded monthly (n=12). We need to find the time (t) in years and months for the initial amount to reach the final amount.

Step 2 ：We use the formula for compound interest, which is \(A = P(1 + \frac{r}{n})^{nt}\).

Step 3 ：Substituting the given values into the formula, we get \(11000 = 8000(1 + \frac{0.074}{12})^{12t}\).

Step 4 ：We can solve this equation for t using logarithms.

Step 5 ：Doing so, we find that t = 4.316684108761635.

Step 6 ：Since t is in years, we convert the decimal part into months by multiplying it by 12. This gives us approximately 4 months.

Step 7 ：Thus, the time required for the initial amount to be at least equal to the final amount is \(\boxed{4}\) years and \(\boxed{4}\) months.