But this takes only into consideration the two selected point used to calculate the the slope and intercept. All other point will not exactly lie on linear function. And as you can choose any combination of two point you will get again infinitely many different parametrization a of the linear model.
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But if the data originates from an exponential, any selection of two points will yield a different slope, because the data point lie not exactly on a straight line.
You suggested to model it linearly, that is what we are discussing here.