I would like to rearrange the logistic function:
$$y=\frac1{1+\exp(-a+bx)}$$
To calculate $x=f(y)$
So I did the following:
$$\frac1{y}=1+\exp(-a+bx)$$ $$\ln\left(\frac1{y}-1\right)=-a+bx$$ $$\frac{\ln\left(\frac1{y}-1\right) +a}{b}=x$$
but when I try to plot this function, it appears that I have lost a '-' somewhere, i.e., I have calculated $-x=f(y)$
Where I have missed the negative sign?