You can earn .54 percent per month at your bank. If you deposit \( \$ 3,700 \), how long must you wait until your account has grown \( \$ 6,700 \) ? Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16 . Number of months

To find out how long you need to wait until your account has grown to $6,700, we can use the formula for compound interest: \[ A = P(1 + r)^n \] where: - \( A \) is the final amount ($6,700), - \( P \) is the principal amount ($3,700), - \( r \) is the monthly interest rate (0.54% or 0.0054), - \( n \) is the number of months. We need to solve for \( n \) when \( A = \$6,700 \) and \( P = \$3,700 \). Substitute the values into the formula: \[ 6700 = 3700(1 + 0.0054)^n \] Now, we can solve for \( n \) using logarithms: \[ \log(6700) = \log(3700) + n \cdot \log(1 + 0.0054) \] \[ n = \frac{\log(6700) - \log(3700)}{\log(1 + 0.0054)} \] Let's calculate the value of \( n \) using logarithms. Calculate the value by following steps: - step0: Calculate: \(\frac{\log_{10}{\left(6700\right)}-\log_{10}{\left(3700\right)}}{\log_{10}{\left(1+0.0054\right)}}\) - step1: Add the numbers: \(\frac{\log_{10}{\left(6700\right)}-\log_{10}{\left(3700\right)}}{\log_{10}{\left(1.0054\right)}}\) - step2: Convert the expressions: \(\frac{\log_{10}{\left(6700\right)}-\log_{10}{\left(3700\right)}}{\log_{10}{\left(\frac{5027}{5000}\right)}}\) - step3: Subtract the terms: \(\frac{\log_{10}{\left(\frac{67}{37}\right)}}{\log_{10}{\left(\frac{5027}{5000}\right)}}\) - step4: Use the logarithm base change rule: \(\log_{\frac{5027}{5000}}{\left(\frac{67}{37}\right)}\) - step5: Rewrite the expression: \(\log_{\frac{5027}{5000}}{\left(67\right)}-\log_{\frac{5027}{5000}}{\left(37\right)}\) The number of months it will take for your account to grow to $6,700 is approximately 110.25 months.