I've read a lot about approximating logarithms recently, and apparently it's not easy. It can be done by Taylor series (slow convergence), by continued fractions (also slow) and also by some limits.
I had an idea to increase the accuracy of a limit approximation by using the Taylor series for $\cosh t$. The limit I got is:
$$\ln x=\lim_{N \to \infty} N \sqrt{2 \left(\sqrt{3} \sqrt{x^{1/N}+x^{-1/N}+1} -3\right)}$$
This limit looks awkward, for sure, but it gives very good approximations for $x$ close to $1$.
For example, even if we take the smallest possible value for $N$, we obtain the value for $\ln 2$ accurate for three digits:
$$N=1,~~~~~x=2$$
$$\sqrt{\sqrt{42}-6}-\ln 2 \approx 0.00021$$
Using the properties of logarithms, we can approximate larger arguments as well, like $\ln 3$:
$$N=1,~~~~~x=\frac{3}{2}$$
$$\sqrt{\sqrt{42}-6}+\sqrt{\sqrt{38}-6}-\ln 3 \approx 0.00022$$
The outline of a proof:
$$\cosh \frac{t}{N}=1+\frac{t^2}{2N^2}+\frac{t^4}{24N^4}+\cdots$$
Now we take only the first three terms and solve the quadratic equation for $t^2=\ln^2 x$.
How does this limit compare to other ways of computing logarithms? I understand that using radicals is not as convenient as rational approximations, but maybe it has some advantages?
