The logistic equation for batch microbial growth is given by the following equation: dtdX=kX(1XmX) a. Use the following approximations: Xm=Yx/sS0,X=Yx/s(S0S) Prove that the logistic equation can be expressed as follows: dtdX=kXS0S b. By using the following relationship, develop an expression describing the variation of substrate concentration with time according to the logistic equation: dtdX=Yx/sdtdS By comparing the form of logistic equation derived in part (a) with the Monod equation at low substrate concentrations (first-order kinetics), determine the Iogistic equation constant (k) in terms of Monod kinetic constants