<Dl> <Dd> l o g 1 p (x) = log ⁡ (1 + x) = 2 a r t a n h (x 2 + x), (\ displaystyle \ mathrm (log1p) (x) = \ log (1 + x) = 2 ~ \ mathrm (artanh) \ left ((\ frac (x) (2 + x)) \ right) \,,) </Dd> </Dl> <Dd> l o g 1 p (x) = log ⁡ (1 + x) = 2 a r t a n h (x 2 + x), (\ displaystyle \ mathrm (log1p) (x) = \ log (1 + x) = 2 ~ \ mathrm (artanh) \ left ((\ frac (x) (2 + x)) \ right) \,,) </Dd> <P> gives a high precision value for small values of x on systems that do not implement log1p (x). </P> <P> The computational complexity of computing the natural logarithm (using the arithmetic - geometric mean) is O (M (n) ln n). Here n is the number of digits of precision at which the natural logarithm is to be evaluated and M (n) is the computational complexity of multiplying two n - digit numbers . </P>

Write the logarithm in terms of natural logarithms