How do you build the logistic model for population growth from the exponential m

The expression for exponential growth is, rmaxN. To obtain the logistic model, this equation is multiplied by 1-N/K. Addition of the term 1-N/K produces sigmoidal growth by simulating environmental resistance to population growth.

Exponential Population Growth Equation : dN/dT = rN,
 where r= intantaneous rate of increase or intrinsic rate of increase. We also can represent it as r = (b -d) where b= per capita birth rate and d= percapita death rate.
 N = population size
But, Logistic Growth Equation is density dependent equation having some assumptions:
1. Unlike exp growth eq, resources for population growth, survival and reproduction are limited.
2. b decreases due to limited resources and intraspecific competition.
3. d increases as population grows due to parasitism, predation, competition etc types of interaction.

Now, model the decreased b : b' = b - aN
 where b = birth rate in uncrowded ideal conditions and a= strength of density dependence, i.e. if a is large, b' drops at each individual added to the population, if a= 0 b'=b, regardless of the N. Larger N, lower the b' and if N come closer to 0, b' tends towards b.
Similarly, model increased d: d' = d+cN

Now, dN/dT= (b'-d')N
=>  dN/dT= [(b-aN)-(d+cN)]N
=>  dN/dT= [(b-d)-(a+c)N]N
=>  dN/dT= {(b-d)/(b-d)} [(b-d)-(a+c)N]N
=>  dN/dT= (b-d)[1-{(a+c)/(b-d)}N]N     {as b-d=r}
=>  dN/dT= rN[1-N/K]

Here, we introduce a new constant K by compiling a,c,b,and d four constants. But it has another biological interpretations. K is defined by the maximum number of the individuals supported in a population, growing according to logistic growth model. It reflects the resource availability of that environment. From the equation, we can establish a relationship between N and K.
1. If N<<K, dN/dT is high, tends towards exp growth.
2. If N>K, dN/dT is negative, population growth is negative and declines towards K.
3. If N=K, dN/dT=0. Population stops to grow. It is denoted as the stable equilibrium.

hence, K is called the carrying capacity

Please, now compare the graphs of both models and make your own decisions.
Ref: A primer to ecology: Gotelli