The amount of money A in an investment after t years with a principal amount P that has a rate of interest of r compounded n times a year is given by,

\(A(t)=P\left(1+\frac{r}{n} \right)^{nt} .\)

Annette want to find the time it will take for an initial investment of $3,000 at 7% interest compounded quarterly to accumulate to $6,500. When she plugs her values into the formula, she get

\(6,500=3,000\left(1+\frac{.7}{4} \right)^{4t} \)

\(\frac{6,500}{3,000} =\left(1+\frac{.7}{4} \right)^{4t}\)

\(\frac{6,500}{3,000} =\left(1.175\right)^{4t} \)

\(\ln \left(\frac{6,500}{3,000} \right)=\ln \left(1.175\right)^{4t} \)

\(\ln \left(\frac{6,500}{3,000} \right)=4t\ln \left(1.175\right)\)

\(\frac{\ln \left(\frac{6,500}{3,000} \right)}{\ln \left(1.175\right)} =4t\)

\(4\left(\frac{\ln \left(\frac{6,500}{3,000} \right)}{\ln \left(1.175\right)} \right)=t\)

4.8=t

The correct answer should be 11.1 years. Explain what went wrong in steps 1-8 in Annette's calculations.