Logarithm is Strictly Concave

Theorem

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

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

Then:

$\ln x: x > 0$ strictly concave.

Proof

From Logarithm is Strictly Increasing, $\ln x$ 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 Real Function is Strictly Concave iff Derivative is Strictly Decreasing, $\ln x$ is strictly concave on $x > 0$.

$\blacksquare$

Sources

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