Of course it is possible to use the Taylor series you mentioned, but it is not very efficient.

Wikipedia lists several other infinite series which converge faster: https://en.wikipedia.org/wiki/Logarithm

@Hanwu Yang: this are in terms of (1+x) or (1-x) . Is it possible in terms of power of x such as log(x)=a0+a1x+a2x^2+---------

I consider it not possible, at least for real number of x - but I am not expert on Mathematics.

If it is possible then Sum(a0,a1,a2,...)=0, because at x=1 log(x)=0;

but a0 is not well defined, as at x=0, log(0)=-infinite.

Not looking at your example but at "... by a convergent infinite series": YES

You may want to check for "CORDIC" (e.g. at wikipedia), which is not the fastest algorithm. But it is reliable and well understood, a lot of "material" being available "out there".