Ln(x) using repeated square root extraction

[Gerson W. Barbosa]

For the sake of completeness here is the recursive limit definition of ln(x), even though this is not quite necessary for our purpose:

\[\ln (x)=\lim_{n\rightarrow \infty} \left [ n\cdot \left ( x^{\frac{1}{n}}-1 \right )-\sum_{k=2}^{\infty }\frac{\ln ^{k}(x)}{k!\cdot n^{k-1}} \right ]\]

If the limit is removed and n is set to 1 then we'll have the following recursive series representation:

\[\ln (x)= \ x-1 -\sum_{k=2}^{\infty }\frac{\ln ^{k}(x)}{k!}\]