Answer
$$35{\text{ years}}$$
Work Step by Step
$$\eqalign{
& P\left( t \right) = 100{e^{t/50}} \cr
& {\text{The initial population is given to }}t = 0 \cr
& P\left( 0 \right) = 100{e^{0/50}} \cr
& P\left( 0 \right) = 100 \cr
& {\text{The double population is }} \cr
& {\text{2}}P\left( 0 \right) = 2\left( {100} \right) \cr
& {\text{2}}P\left( 0 \right) = 200 \cr
& {\text{Then to reach 200 the time is}} \cr
& 200 = 100{e^{t/50}} \cr
& {\text{Solve for }}t \cr
& \frac{{200}}{{100}} = {e^{t/50}} \cr
& 2 = {e^{t/50}} \cr
& \ln \left( 2 \right) = \ln {e^{t/50}} \cr
& \ln \left( 2 \right) = \frac{t}{{50}} \cr
& t = 50\ln \left( 2 \right) \cr
& t = 34.65735903 \cr
& t \approx 35{\text{ years}} \cr} $$
