# What are the differences between Logistic Function and Sigmoid Function?

Fig 1. Logistic Function

Fig 2. Sigmoid Function

is it more like generalized kind of sigmoid function where you could have a higher maximum value?

• $S$ is an increasing function in $t$, $f$ is increasing for values of $x \leq x_0$, decreasing for values of $x \geq x_0$ – mic Mar 30 '16 at 7:25

Yes, the sigmoid function is a special case of the Logistic function when $L=1$, $k=1$, $x_0 =0$.

If you play around with the parameters (Wolfram Alpha), you will see that

• $L$ is the maximum value the function can take. $e^{-k(x-x_0)}$ is always greater or equal than 0, so the maximum point is achieved when it it 0, and is at $L/1$.

• $x_0$ controls where on the $x$ axis the growth should the, because if you put $x_0$ in the function, $x_0 - x_0$ cancel out and $e^0 = 1$, so you end up with $f(x_0) = L/2$, the midpoint of the growth.

• the parameter $k$ controls how steep the change from the minimum to the maximum value is.

The logistic function is: $$f(x) = \frac{K}{1+Ce^{-rx}}$$ where $$C$$ is the constant from integration, $$r$$ is the proportionality constant, and $$K$$ is the threshold limit.

Assuming the limits are between $$0$$ and $$1$$, we get $$\frac{1}{1+e^{-x}}$$ which is the sigmoid function.