# A fast converging limit for $\ln x$ (or why $\ln 2 \approx \sqrt{\sqrt{42}-6})$

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?

• This series is very efficient and also gives rational approximations: math.stackexchange.com/a/61283/155629 – Travis Apr 27 '16 at 10:02
• @Travis, you are right, this series is much better, it gives a better approxiamtion for $3$ terms already. – Yuriy S Apr 27 '16 at 11:27