Answer
$\text{approximately }34.7\text{ years}$
Work Step by Step
Since the amount needs to double, then $A=10,000$. Substituting the known values in the formula $A=P\left( 1+\dfrac{r}{n} \right)^{nt}$, then,
\begin{array}{l}
10,000=5,000\left( 1+\dfrac{0.02}{4} \right)^{4t}
\\\\
2=\left( 1+\dfrac{0.02}{4} \right)^{4t}
\text{...divide both sides by $5,000$}
\\\\
2=\left( 1+0.005 \right)^{4t}
\\
2=\left( 1.005 \right)^{4t}
\\
\log2=\log\left( 1.005 \right)^{4t}
\text{...get the logarithm of both sides}
\\
\log2=4t(\log1.005)
\\
\dfrac{\log2}{4(\log1.005)}=t
\\
t\approx34.7
.\end{array}
Hence, the money will double in $
\text{approximately }34.7\text{ years}
.$