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 $. - Abu Waleed Tea