Exponential and Logarithmic Functions
Finding the time in a word problem on compound interest

Suppose that $\$ 2000$ is placed in a savings account at an annual rate of $6 \%$, compounded monthly. Assuming that no withdrawals are made, how long will it take for the account to grow to $\$ 3212$ ?

Do not round any intermediate computations, and round your answer to the nearest hundredth.
years

Step 1 ：Given that the principal amount \(P = \$2000\), the annual interest rate \(r = 6\% = 0.06\), the number of times interest is compounded per year \(n = 12\), and the final amount \(A = \$3212\).

Step 2 ：We want to find the time \(t\) it will take for the account to grow to \$3212.

Step 3 ：We can use the formula for compound interest, which is \(A = P(1 + r/n)^{nt}\).

Step 4 ：Rearranging the formula to solve for \(t\), we get \(t = \frac{\log(A/P)}{n \log(1 + r/n)}\).

Step 5 ：Substituting the given values into the formula, we get \(t = \frac{\log(3212/2000)}{12 \log(1 + 0.06/12)}\).

Step 6 ：Calculating the above expression, we find that \(t \approx 7.92\).

Step 7 ：So, the time it will take for the account to grow to \$3212 is approximately \(\boxed{7.92}\) years.