When does function $(\log_b(x))^p$ change its curvature?

Question

Consider $(\log_b(x))^p$ where $b$ is a constant $>1$; $x, p \in \mathbb R_+$.

As we increase the value of $p$ (starting from 1), at specific value of $p$, the curve changes its shape from concave to convex, specifically for $x\ge 1$. See Curve Transition.

So the question is:

1. At what value of $p$, in terms of $b$, the transition of curvature (from concave to convex) occurs?
2. How to find it mathematically?

Answer 1

As we increase the value of $p$ (starting from $1$), at specific value of $p$, the curve changes its shape from concave to convex

No, it only seems that way, because of the restricted window in which you see the graph. The function $(\log_b x)^p$ is not convex for any value $p>0$.

It suffices to consider $\ln x$, because the base only contributes a constant factor $\ln b$ to the denominator. The second derivative of $(\ln x)^p$ is $$\frac{p(p-\ln x-1)(\ln x)^{p-2}}{x^2}$$

• If $0<p\le 1$, the second derivative is negative for all $x>1$, so the function is concave
• If $p>1$, the second derivative changes sign at the point where $\ln x = p-1$. This inflection point will be far off to the right when $p$ is large, but it's there.

That said, the function is convex on $(1,\infty)$ when $p<0$.