# What is the difference between logarithmic decay vs exponential decay? I

What is the difference between logarithmic decay vs exponential decay?
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.
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toriannucz
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 $f\left(x\right)=A{e}^{kx}+C$ 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 $g\left(x\right)=A\mathrm{ln}\left(x\right)+C$ 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 $\mathrm{ln}\left(x\right)$ aren't even defined on half the real line, whereas the exponential ones are defined everywhere.

letumsnemesislh
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.