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