Using either logarithms or a graphing calculator, find the time required for the initial amount to be at least equal to the final amount. $\$ 4700$, deposited at 8.4\% compounded monthly, to reach at least $\$ 9400$
The time required is $\square$ year(s) and $\square$ months.

Step 1 ：Given that the initial amount (P) is $4700, the final amount (A) is $9400, the annual interest rate (r) is 8.4% or 0.084, and the interest is compounded monthly (n=12 times per year), we need to find the time (t) in years and months.

Step 2 ：We 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: \(t = \frac{\ln(\frac{A}{P})}{n \cdot \ln(1 + \frac{r}{n})}\).

Step 4 ：Substituting the given values into the formula, we get \(t = \frac{\ln(\frac{9400}{4700})}{12 \cdot \ln(1 + \frac{0.084}{12})}\).

Step 5 ：Solving the equation, we get t = 8.28059970480305.

Step 6 ：Since t is in years, we convert the decimal part into months by multiplying it by 12. This gives us approximately 3 months.

Step 7 ：Final Answer: The time required is \(\boxed{8}\) year(s) and \(\boxed{3}\) months.