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I am a little unclear on whether they are distinctly different or whether this is a 'square is a rectangle, but rectangle is not necessarily a square' type of relationship.

jterm
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    I've never heard the terms "logarithmic decay" and "exponential decay". Could you provide some context, maybe a link? – Jack M Nov 08 '13 at 18:09
  • "exponential decay" describes things that have a half-life and is a very common term. I'm not sure what "logarithmic decay" means, if anything. – Jeff Learman Jan 27 '23 at 20:59
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    For whimsy, I think log decay should look like log growth, but running time backwards. Imagine flipping the graph horizontally. Whimsy over :( Its not a thing, because it reduces to the complement of the exponential. Imagine a falling object, acceleration is constant, velocity increases linearly (until terminal velocity or splat). Distance to the ground is decaying slowly and then faster, you might call it logarithmic, for whimsy, but your distance traveled is the complement of that, and it grows exponentially so is enough to model it that way. You don't need to model both sides of the mirror. – Davos Jun 11 '25 at 17:35

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The natural logarithm and exponential are inverses of one another, so the associated slopes will also be inverses. If you put exponentially decaying data on a log plot, i.e. log of the exponential decaying data with the same input, you get a linear plot. If you put the logarithmic decaying plot on an exponential plot (exponential of the data), you get a linear plot, so the way they are decaying is exactly opposite.

Jeremy Upsal
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  • I thought this was the case but came here to check. I wasn't crazy about your explanation though, for someone new or very rusty it doesn't really make the relationship clear. Also 5 people ticked the other answer… so I'm hoping a third answer gets written. – wide_eyed_pupil Aug 16 '19 at 00:45
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The "Square is a rectangle" relationship is an example where the square is a special case of a rectangle.

"Exponential decay" gets its name because the functions used to model it are of the form $f(x)=Ae^{kx} +C$ where $A>0$ and $k<0$. (Other $k$'s above $0$ yield an increasing function, not a decaying one.)

Similarly for "logarithmic decay," it gets its name since its modeled with functions of the form $g(x)=A\ln(x)+C$ where $A<0$.

These two families of functions do not overlap, so neither is a special case of the other. The giveaway is that the functions with $\ln(x)$ aren't even defined on half the real line, whereas the exponential ones are defined everywhere.

rschwieb
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  • +1. thanks. could you also explain what is polynomial decay and polynomial growth? Is polynomial decay $x^{-a}$ and polynomial growth $x^a$, where $a>0$? – Tim Apr 30 '14 at 02:23
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    This is correct for exponential decay. I don't think there is such a thing as logarithmic decay. Note that while exponential decay as stated above has an asymptote, logarithmic decay does not -- it decreases without limit. Which makes it not "decay." – Jeff Learman Jan 27 '23 at 21:12
  • I think it makes sense to envision a decay that is "slower" than exponential, or a decay that is "slow as a logarithm". But this is certainly not the functional form for that. – deps_stats Mar 22 '24 at 14:54
  • @JeffLearman I think there's no need for logarithmic decay because (naively) it would be a mirror flip of the exponential. I am not sure why decreasing without limit vs being asymptotic would preclude it from being "decay". An abstract function can be continuous beyond the range where it could meaningfully map to real world processes that are called "decay", like radioactive decay or rotting compost decay. Sure those can't reduce below zero, is that what you mean? – Davos Jun 11 '25 at 17:44
  • @davos - yes, except that we can "decay" to a nonzero constant, which is what an RC filter can do in certain situations. – Jeff Learman Jun 19 '25 at 16:55
  • On further thought, I withdraw my assertion that there's no such thing as logarithmic decay. My main argument against it also applies to linear decay, which is assumed to stop at zero (or any constant value), so no reason logarithmic decay can't work the same way. But I doubt it's a useful concept, and I suspect anytime someone mentions "logarithmic decay" they actually mean "exponential decay." In fact, if I google it, I get the definition of exponential decay, labeled as logarithmic decay. – Jeff Learman Jun 19 '25 at 21:19
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Experiment with it by entering ln and exp in this online grapher: https://mathopenref.com/graphfunctions.html exp and ln graphs, how they are related and the influence of a vs -a Image: https://i.sstatic.net/W4cYw.jpg At school, we learn about exponentials being typically at how powerful exponential growth rates are e.g. birth rates reaching over population or growth of bacteria

e.g. h(x)=a * exp(x)+c

Log is rather indicating how it can take forever to reach certain limit, e.g. why - IF you take anti-biotics to kill a bacteria infection - you have to take them so long because you're always killing e.g. half of them the first day, then again half of that remaining half, etc. It explains why you feel good, so quickly. And also why you mustn't stop. Because in the remaining half or quarter, there's plenty of bacteria still and if just 1 manages to find a way around the anti-biotic thanks to a mutation, it will grow exponentially and make you sick again in a matter of days. And then you won't have any anti-biotic anymore to kill it off to zero = to the last single bacteria and keep taking the anti-biotic for a couple of days more, just to make sure you have killed them all.

e.g. f(x)=a * ln(x)+c

And then we learn how they are both linked and can be flipped, and pivoted by e.g. turning the a into -a.

Try it out.

Conclusion: exponential or logarithmic decay? An exponential decay can drill down under zero, a logarithmic decay can reach zero-eventually but not down under zero?

Sven AERTS
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  • There's a mistake here somewhere, because exponential decay is equivalent to anything that has a half-life. You never get to zero unless you're dealing with discrete values (e.g., the number of bacteria.) Exponential decay is what you get from a step input into an RC filter. – Jeff Learman Jan 27 '23 at 20:53