Real Natural Logarithm is Restriction of Complex Natural Logarithm

Theorem
Let $\ln: \C_{\ne 0} \to \C$ be the complex natural logarithm.

Let $\ln': \R_{>0} \to \R$ be the real natural logarithm.

Then:
 * $\ln' = \ln \restriction_{\R_{>0} \times \R}$

That is, the real natural logarithm is the restriction of the complex natural logarithm.

Proof
From Domain of Real Natural Logarithm:
 * $\operatorname{Dom} \left({\ln'}\right) = \R_{>0}$

From Image of Real Natural Logarithm:
 * $\operatorname{Im} \left({\ln'}\right) = \R$

Let $z \in \C$ such that $z = x + i y$.

Let $z$ be expressed in exponential form as $z = r e^{i \theta}$.

Let $x > 0$ and $y = 0$.

Thus $z \in \R_{>0}$.

Then:

In order for $\ln z \in \R$ it is necessary that $k = 0$.

Thus $\ln z = \ln' x$.

Hence the result.