Solution: Given $ R(4) = 2R(0) $, substitute into the equation: $ 1000e^{4k} = 2 \cdot 1000 $. Simplify to $ e^{4k} = 2 $. Take the natural logarithm: $ 4k = \ln(2) $, so $ k = rac{\ln(2)}{4} $. Thus, $ oxed{\dfrac{\ln(2)}{4}} $.