Most are familiar with the term linear regression which, in simple terms, attempts to model the (linear) relationship between two variables (assuming there is one) by fitting a best-fit linear equation (line) to a set of observed data. The most common method for fitting a regression line is the method of least-squares. The reader is encouraged to reference the paper, Mathematics of Least Squares, which presents the mathematics of least squares in detail, inclusive of the calculus and the linear algebra approaches, and Linear Regression which details an interactive application developed to explore linear regression (and provides more detailed information such as the Pearson Correlation Coefficient).
This paper will explore the mathematics of perhaps one of the simplest nonlinear regression models, exponential regression. Exponential regression, in simple terms, attempts to fit the best exponential function to a given set of data which exhibit exponential behaviour (characteristics). That is, to determine the equation of an exponential function which bests models the data.
Mathematics of Exponential Regression