I am looking for the correct terminology to describe the region of greatest "curve", or "bend" in a logarithmic function. Because of the nature of logarithmic functions I don't think this meets the definition of a "reversal point" or "curve reversal". Specifically, due to the system that I'm describing, I'd like to draw a differentiation between the "stable" portion of the function (in the context I'm describing) that primarily runs across the X axis, and the period of steep decay that runs across the y-axis. I hope to do this by describing the rapidity of the transition between these two states in terms of the "curve" between. Is there an existing nomenclature here?

  • $\begingroup$ I don't think there is such a term. You want something like "inflection point" but that already means something else. $\endgroup$ – kimchi lover Sep 17 '17 at 17:45
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    $\begingroup$ The logarithm function that I know is monotonically increasing, there is nothing that seems to be a "reversal point". Its increases slows but gradually, there is no sudden transition from fast to slow. $\endgroup$ – badjohn Sep 17 '17 at 17:46
  • $\begingroup$ $x=1$ would be natural for your purposes. $\endgroup$ – MathematicsStudent1122 Sep 17 '17 at 17:49
  • $\begingroup$ Do you mean a "logistic" function? $\endgroup$ – Henning Makholm Sep 17 '17 at 17:49
  • $\begingroup$ I don't, I think, but it's worth discussion: I mean exponential, and bounded exponential, but not the amalgam, which I would term logistic. There is an interesting side question as to whether the term 'logarithmic' subsumes the term 'bounded logarithmic'. Do you think my terms are in order? $\endgroup$ – Industrademic Sep 21 '17 at 2:03

For a function $y$ of a variable $x$, the curvature of the graph of $y$ is $$ \kappa = \frac{y''}{(1 + y'^2)^\frac{3}{2}} $$ which, when you apply it to $y = \ln x$, gives

$$ \kappa = \frac{\frac{-1}{x^2}}{(1 + (1/x)^2)^\frac{3}{2}} $$ Multiplying top and bottom by $x^3$ gives \begin{align} \kappa &= \frac{-x}{x^3(1 + (1/x)^2)^\frac{3}{2}}\\ &= \frac{-x}{(x^2)^\frac{3}{2}(1 + (1/x)^2)^\frac{3}{2}}\\ &= \frac{-x}{(x^2 + 1)^\frac{3}{2}} \end{align} As $x$ gets large, this goes to zero. As $x$ goes to $0$, it approaches $0$ as well. So it has a critical point somewhere in between.

Graphing suggests that the critical point occurs at $x = \frac{\sqrt{2}}{2}$, but you can use some calculus to determine that directly.

To answer the second question: I don't think that there's an accepted term for this, although sometimes folks talk about the "knee" of a curve, and maybe that captures what you're thinking of.

  • $\begingroup$ Curvature = 1/radius of curvature, I think I recall, so maybe turning point or pivot point would be appropriate things to call it. With an explanation for this nonce term, of course. $\endgroup$ – kimchi lover Sep 17 '17 at 18:10
  • $\begingroup$ The name is actually the only question. "Knee" is interesting, but on some log functions is not a great descriptor... "Pivot point" is interesting. Possibly most interesting is that there is indeed no accepted term. Everyone in my lab was sure there must be, and that we had simply forgotten it... $\endgroup$ – Industrademic Sep 18 '17 at 14:02
  • $\begingroup$ Sorry for wasting your time; I thought it might be useful to check that in a log function there actually WAS a point of greatest curvature, so that we weren't searching for a name for a thing that didn't exist. $\endgroup$ – John Hughes Sep 18 '17 at 16:56
  • $\begingroup$ You in no way wasted my time, and your response is interesting and relevant. Thank you for it. $\endgroup$ – Industrademic Sep 21 '17 at 2:00

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