# What is exponential decay, exactly?

On the web (ie, Wikipedia, and other sites) it seems that exponential decay is always defined as the situation $f\;'(t) = -kf(t)$. However, is not Newton’s Law of Cooling an example of exponential decay? But Newton’s Law of Cooling does not fit this form. What is missing is a constant. If we define exponential decay to be the situation $f'(t) = w - kf(t)$, then everything’s fine. So, is there in fact a constant missing (for convenience?) from the widely-disseminated definition of exponential decay, or, as I suspect, is the definition not all that fixed, depending on some judicious arm-waving as the occasion demands?

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If you say that $f(t)$ is the difference, at time $t$, between the object's temperature and the temperature of the surroundings, then Newton's law of cooling does fit that form. – Michael Hardy Oct 19 '11 at 2:05
@Michael Hardy: Bingo. THAT answers my question. I'm shamefaced I didn't notice it myself. Anyway, if you will post your comment as an answer, I will accept it. – Mike Jones Oct 19 '11 at 22:13

With a positive constant $k,$ and other constants $A,B,$ exponential decay is $$f(t) = A + B e^{- k t},$$ where you can work out the O.D.E. as you like. Examples include the battery charging curve, also called capacitor, in which $A$ is positive and $B= -A.$
Solutions to $f'=-kf$ decay (exponentially) to zero. Your cup of coffee doesn't cool down to zero degrees, it cools down (exponentially) to room temperature, and that's the $w$ of $f'=w-kf$. If you rejigged your temperature scale to measure degrees above room temperature instead of degrees above zero, Newton Cooling would be $f'=-kf$.