Logarithm is Strictly Increasing and Concave

Theorem

Let $x \in \R$ be a real number such that $x > 0$.

Let $\ln x$ be the natural logarithm of $x$.

Then:

$\ln x: x > 0$ is strictly increasing and concave.

Proof

From Derivative of Natural Logarithm Function $D \ln x = \dfrac 1 x$, which is strictly positive on $x > 0$.

From Derivative of Monotone Function it follows that $\ln x$ is strictly increasing on $x > 0$.

From Second Derivative of Natural Logarithm Function:

$D^2 \ln x = -\dfrac 1 {x^2}$

Thus $D^2 \ln x$ is strictly negative on $x > 0$ (in fact is strictly negative for all $x \ne 0$).

Thus from Derivative of Monotone Function, $D \dfrac 1 x$ is strictly decreasing on $x > 0$.

So from Derivative of Convex or Concave Function, $\ln x$ is concave on $x > 0$.

$\blacksquare$

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 14.1$