# What do you call the reversal point of a logarithm?

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?

• I don't think there is such a term. You want something like "inflection point" but that already means something else. – kimchi lover Sep 17 '17 at 17:45
• 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. – badjohn Sep 17 '17 at 17:46
• $x=1$ would be natural for your purposes. – MathematicsStudent1122 Sep 17 '17 at 17:49
• Do you mean a "logistic" function? – Henning Makholm Sep 17 '17 at 17:49
• 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? – 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.