$\epsilon$-$\delta$ proof that $\ln x$ is continuous everywhere on its domain

Question

How to use $\epsilon$-$\delta$ argument to show the global continuity of the function $x \mapsto \ln x$ on its domain?

For, if $c$ lies in the domain of $\ln x$ then $|\ln x - \ln c| = |\ln \frac{x}{c}|$. So how to relate the distance between the functional values to the distance between the arguments?

Answer 1

$x-c<\delta\Rightarrow x<c(\delta c^{-1}+1)\Rightarrow \ln x<\ln c + \ln(\delta c^{-1}+1)\Rightarrow \ln x-\ln c < \ln(\delta c^{-1}+1)$

Does this make some sense??