# Real Natural Logarithm Function is Continuous

## Proof 1

We have that the Natural Logarithm Function is Differentiable.

The result follows from Differentiable Function is Continuous.

$\blacksquare$

## Proof 2

$\dfrac 1 2 < \map \ln 2 < 1$

Fix $x \in \R$.

Consider $\dfrac x {\map \ln 2}$.

$\forall \epsilon \in \R_{>0} \exists r \in \Q : \size {r - \dfrac x {\map \ln 2} } < \epsilon$

Thus:

 $\ds \size {r - \dfrac x {\map \ln 2} }$ $<$ $\ds \epsilon$ $\ds \leadsto \ \$ $\ds \map \ln 2 \size {r - \dfrac x {\map \ln 2} }$ $=$ $\ds \size {\map \ln {2^r} - x }$ Natural Logarithm of Rational Power $\ds$ $<$ $\ds \epsilon \, \map \ln 2$ Real Number Ordering is Compatible with Multiplication $\ds$ $<$ $\ds \epsilon$ as $\map \ln 2 < 1$ $\ds \leadsto \ \$ $\ds \size {\map \ln t - x}$ $<$ $\ds \epsilon$ substituting $t = 2^r$

Thus:

$\forall \epsilon \in \R_{>0}: \exists t \in \R_{>0}: \size {\map \ln t - x} < \epsilon$

Thus, the image of $\R_{>0}$ under $\ln$ is everywhere dense in $\R$.

From Monotone Real Function with Everywhere Dense Image is Continuous, $\ln$ is continuous on $\R_{>0}$.

Hence the result.

$\blacksquare$