[tex]~~~~~~ \textit{Compound Interest Earned Amount}\\\\A=P\left(1+\frac{r}{n}\right)^{nt}\quad\begin{cases}A=\textit{accumulated amount}\dotfill &\$2900\\P=\textit{original amount deposited}\dotfill &\$2000\\r=rate\to 6.75\%\to \frac{6.75}{100}\dotfill &0.0675\\n=\begin{array}{llll}\textit{times it compounds per year}\\\textit{monthly, thus twelve}\end{array}\dotfill &12\\t=years\end{cases}[/tex]
[tex]2900=2000\left(1+\frac{0.0675}{12}\right)^{12\cdot t}\implies \cfrac{2900}{2000}=\left(1+\frac{0.0675}{12}\right)^{12t} \\\\\\ \cfrac{29}{20}=(1.005625)^{12t}\implies \log\left( \cfrac{29}{20} \right)=\log(1.005625^{12t}) \\\\\\ \log\left( \cfrac{29}{20} \right)=t\log(1.005625^{12})\implies \cfrac{\log\left( \frac{29}{20} \right)}{\log(1.005625^{12})}=t\implies \stackrel{years}{5.5\approx t}[/tex]
