Analytical Method to Find a Point on the Best-Fitting Curve

Remember, that our exponential curve is in this form,

y = a e^(bx)

Let's take the natural log of both sides of the equation.

  1. Ln[y] = Ln[a e^(bx)]

Recall the properties of logarithms that state

Ln[a b] = Ln[a] + Ln[b], and ,
Ln[e^x] = x.

Therefore, from equation (1) we get,

Ln[y] = Ln[a] + Ln[e^(bx)]

Ln[y] = Ln[a] + b x

Ln[y] = b x + Ln[a].

This looks like the equation of a line,

Ln[y] = b x + Ln[a]

(y) = (m x) + (b).

Assuming that the best fitting line goes through the point (xmean, ymean) for a line, we can show that it is the same point for an exponential except we use the mean of Ln[y] instead of y.

The mean data point of this equation would be the mean of x and the mean of Ln[y]. However, this is for a line and not for the curve. We are interested in this value in terms of y. To find this we need to raise e to the mean of Ln[y] to get back to the curve. Therefore, the point that must be on the best fitting curve of an exponential function is the point, (xmean, e ^ (mean of Ln[y])).