How long will it take $\$ 5,000$ to grow to $\$ 9,000$ if it is invested at $7 \%$ compounded monthly?

Step 1 ：Given that the principal amount (P) is $5000, the final amount (A) is $9000, the annual interest rate (r) is 7% or 0.07 in decimal form, and the interest is compounded monthly (n=12). We are asked to find the time (t) it will take for the principal amount to grow to the final amount.

Step 2 ：Using the formula for compound interest, A = P(1 + r/n)^(nt), we can rearrange the formula to solve for t: t = log(A/P) / (n * log(1 + r/n)).

Step 3 ：Substituting the given values into the formula, we get t = log(9000/5000) / (12 * log(1 + 0.07/12)).

Step 4 ：Calculating the above expression, we find that t ≈ 8.42.

Step 5 ：So, it will take approximately \(\boxed{8.42}\) years for $5000 to grow to $9000 if it is invested at 7% compounded monthly.