Question

Find the time required for $5000 to be equal to $14,000 when deposited at 7% compounded monthly

Answer 1

14.7 years will give you $13,998.33 that should be close enough.
See the attached formula.

Answer 2

`\bf \qquad \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\to &\$14000\\ P=\textit{original amount deposited}\to &\$5000\\ r=rate\to 7\%\to (7)/(100)\to &0.07\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\to &12\\ t=years \end{cases} \\\\\\ 14000=5000\left(1+(0.07)/(12)\right)^(12t)\implies \cfrac{14000}{5000}=\left(1+(0.07)/(12)\right)^(12t) \\\\\\`


`\bf \cfrac{14}{5}=\left( 1+(7)/(1200) \right)^(12t)\implies \cfrac{14}{5}=\left(\cfrac{1207}{1200} \right)^(12t) \\\\\\ log\left((14)/(5) \right)=log\left[ \left((1207)/(1200) \right)^(12t) \right]\implies log\left((14)/(5) \right)=12t\cdot log\left[ \left((1207)/(1200) \right) \right] \\\\\\ \cfrac{log\left((14)/(5) \right)}{12\cdot log\left[ \left((1207)/(1200) \right) \right]}=t\implies 14.75\approx t`

so, about 14 years and 9 months.