# How to find an inflection point (maximum slope) of a log-log plot?

I have a function $f(x)$ and I have plotted it on a log-log plot. I would like to find the location of the inflection point of the function. (note: that means the inflection point in "log-log space" which is not the same as the inflection point of $f(x)$ itself.) I know the following is true: $\frac {d \log y}{d \log x} = \frac{x}{y} \frac{dy}{dx}$. What is the correct formula to find the local maximum of $\frac {d \log y}{d \log x}$?

• I believe I got the correct answer by taking the derivative of $\frac{x}{y} \frac{dy}{dx}$, and finding the local maximum by setting the result equal to zero and solving for $x$. Is that the correct approach? – qdread Jun 15 '18 at 17:53

Suppose $y=f(x)$ for some function $f$. When you make a log-log plot you are essentially introducing two new variables $$u = \log x, v = \log y$$ and plotting the function $$v=g(u)$$ You can express $g$ in terms of $f$ as follows: $$v = \log y = \log f(x) = \log f(e^u)$$ and therefore the function you are interested in is $g(u) = \log f(e^u)$.
Now, you are looking for the inflection point(s) of $g$. That means you need to study the second derivative of $g$, i.e. $g''(u) = \frac{d^2v}{du^2}$. By the chain rule $$g'(u) = \frac{f'(e^u)e^u}{f(e^u)}$$ and then using the quotient rule and chain rule $$g''(u) =\frac{f(e^u)\left(f''(e^u)e^{2u}+f'(e^u)e^u\right)-\left(f'(e^u)\right)^2e^{2u}}{(f(e^u)^2}$$
In terms of the original variables $x$ and $y$, this is $$\frac{y\left(f''(x)x^{2}+f'(x)x\right)-\left(f'(x)\right)^2x^{2}}{y^2}$$
The denominator will be zero if $y=0$, but since you are doing a log-log plot presumably you know that $y>0$ so we can discard this case. The numerator will be zero if $$y\left(f''(x)x^{2}+f'(x)x\right)=\left(f'(x)\right)^2x^{2}$$ Once again since we are doing a log-log plot I assume you already know that $x>0$, so we can divide through by $x^2$ to get $$y\left(f''(x)+\frac{f'(x)}{x}\right)=\left(f'(x)\right)^2$$
The criterion $g''(u)=0$ is equivalent to this equation relating $f(x)$ and its derivatives $f'(x)$ and $f(x)$. I don't see any simpler way to write this but perhaps if you know something about $f(x)$ you might be able to simplify it still further.