# Natural Logarithm Function is Continuous

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## 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:

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

$\blacksquare$