Property 1: Given samples {x₁, …, x_n} and {y₁, …, y_n} and let ŷ = αe^βx, then the value of α and β that minimize (y_i − ŷ_i)² satisfy the following equations:

Proof: The minimum is obtained when the first partial derivatives are 0. Let

Thus we seek values for α and β such that  and ; i.e.

Property 2: Under the same assumptions as Property 1, given initial guesses α₀ and β₀ forα  and β, let F = [f  g]^T where f and g are as in Property 1 and

Now define the 2 × 1 column vectors B_n and the 2 × 2 matrices J_n  recursively as follows

Then provided α₀ and β₀ are sufficiently close to the coefficient values that minimize the sum of the deviations squared, then B_n converges to such coefficient values.

Proof: Now

Thus

The proof now follows by Property 2 of Newton’s Method.