How can we determine the value of A, B and C in the logistic function: f(x) = A / (1 + Be^(–Cx))? Consider we have the values of f(x) and x.

sciencesolve | Certified Educator

The parameter A is the limiting value of the data.

You may find the values of the parameters B and C using the properties of exponential functions. The parameters B and C are the y intercepts of the function f(x).

Baesd on the properties of exponential function, the base C must be positive.

If C>0 => `e^(-Cx)=1/e^(Cx)`  => the values of `e^(-Cx) ` decrease.

If C<0 => the values of `e^(-Cx)`  increase.

To find A,B,C, use the fact that the logistic function f(x) resembles with the exponential function.

In the early stages of growth, the function f(x) behaves like `f(x)= (A/B)*e^(Cx).`

`f(x)= ((A/B)*e^(C(x + Delta x)))/((A/B)*e^(Cx))`

Reduce:

`f(x)=(e^(C*Delta x))`

Put ``and calculate C.

Let say that the ratio  `e^(C*Delta x)= 2.5`

2.5 = `e^(C*Delta x)`

Take logarithms:

ln 2.5 = `ln e^C*Delta x`

Use the property of logarithms `ln a^b`  = `b*ln a`

`ln e^(C*Delta x) = C*Delta x*ln e`

Put ln e=1 and `Delta x = 10`

ln 2.5=C*10

`C = ln 2.5/10`

In the lately stages of growth, the function f(x) behaves like `f(x)= e^(-C*Delta x).`

Put `Delta x=10` and calculate C.

Let say that the ratio  `e^(-C*Delta x)=0.5.`

`0.5 = e^(-C*Delta x)`

Take logarithms:

`ln 0.5 = ln e^(-C*Delta x)`

Use the property of logarithms `ln a^b = b*ln a`

`ln e^(-C*Delta x) = -C*Delta x*ln e`

Put ln e=1 and `Delta x = 10`

`ln 0.5=-C*10`

`C = -ln 0.5/10`

Answer: The values of A and C are A=10 and  C = ln 2.5/10 or C = ln 0.5/10. B depends on A and B.