Answer
$$451{\text{ years}}$$
Work Step by Step
$$\eqalign{
& m\left( t \right) = 100{e^{ - t/650}} \cr
& m = 50{\text{grams}} \cr
& {\text{Let }}m\left( t \right) = 50 \cr
& 50 = 100{e^{ - t/650}} \cr
& {\text{Solve for }}t \cr
& \frac{{50}}{{100}} = {e^{ - t/650}} \cr
& 0.5 = {e^{ - t/650}} \cr
& \ln \left( {0.5} \right) = \ln {e^{ - t/650}} \cr
& \ln \left( {0.5} \right) = - \frac{t}{{650}} \cr
& t = - 650\ln \left( {0.5} \right) \cr
& t = 450.5456674 \cr
& t \approx 451{\text{ years}} \cr} $$
