# Generating Machin Type formulas with inverse hyperbolic tangents for logarithms

Machin Type formulas for $$\pi$$ have the following general form: $$c_{0} \frac{\pi}{4}=\sum_{n=1}^{N} c_{n} \arctan \frac{a_{n}}{b_{n}}$$ Recently browsing through this question here, I really became very curious in finding out Machin Type formula's for logarithms,of course using the inverse hyperbolic tangent function.

Now, its relatively easy and well known to find out Machin Type formulas for $$\pi$$ using complex numbers the original equation can be written as $$(1+i)^{c_{0}}=\prod_{n=1}^{N}\left(b_{n}+a_{n} i\right)^{c_{n}}$$ and then suitable $$a_n$$ and $$b_n$$ could be found using algorithms like branch and bound search, with arbitrary small $$\frac{a_n}{b_n}$$ for efficient computation.

Now what I am really curious about is the find whether there are methods for obtaining such a series for $$\log k$$ which could be in the form of $$c_{0} \log k=\sum_{n=1}^{N} c_{n} \tanh ^{-1}\frac{a_{n}}{b_{n}}$$ For arbitary $$k$$?

$$\log(x) = -2 \tanh^{-1} \left( \frac{1-x}{1+x} \right)\ \text{for}\ x > 0$$ So just write $$k = \prod_j x_j^{e_j}$$ with $$x_j > 0$$, and then $$\log(k) = \sum_{j} -2 e_j \tanh^{-1}\left(\frac{1-x_j}{1+x_j}\right)$$
• This is pretty good,but the question is what is one wants may be more efficiency for computation (ie. $$\sum_{n=1}^N \log(\frac{b_n}{a_n})$$ is minimum. Commented Mar 19, 2019 at 15:40