I am trying to find the order of logarithmic expressions as $x \rightarrow 0$. For example I can find that $\ln(1+x) = \mathcal{O}(x)$ and $\ln(1+x) = \mathcal{o}(1)$.
But when dealing with more complicated expressions below:
$\ln\left[ 1+\frac{\ln(1+2x)}{1-2x} \right]$ and $\ln\left[ 1+\frac{1+2x}{x(1-2x)} \right].$
I can find $\ln\left[ 1+\frac{\ln(1+2x)}{1-2x} \right]=\mathcal{o}(1)$, but how would I determine $\mathcal{O}$, and also for the 2nd expression?