Solve the introductory proportional-decay equation

For $N\ne0$, $\ln|N|=-kt+C$, so $N=Ae^{-kt}$ with nonzero A. The missing case $N=0$ also solves the equation, so A may be any real constant. For a positive physical amount, $A=N(0)>0$, and the amount approaches zero without reaching it at finite time in this continuous model.