# Logarithm is Strictly Increasing

## 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.

### Corollary

Let $\ln$ be the natural logarithm.

Then $\ln$ is injective on $\R_{>0}$.

## 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 $x > 0$.

$\blacksquare$