Logarithm is Strictly Increasing

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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 $x > 0$.

$\blacksquare$


Sources