Logrithm - Calculation - Power Series

Power Series

Taylor series

For any real number z that satisfies 0 < z < 2, the following formula holds:

This is a shorthand for saying that ln(z) can be approximated to a more and more accurate value by the following expressions:

For example, with z = 1.5 the third approximation yields 0.4167, which is about 0.011 greater than ln(1.5) = 0.405465. This series approximates ln(z) with arbitrary precision, provided the number of summands is large enough. In elementary calculus, ln(z) is therefore the limit of this series. It is the Taylor series of the natural logarithm at z = 1. The Taylor series of ln z provides a particularly useful approximation to ln(1+z) when z is small, |z| << 1, since then

For example, with z = 0.1 the first-order approximation gives ln(1.1) ≈ 0.1, which is less than 5% off the correct value 0.0953.

More efficient series

Another series is based on the area hyperbolic tangent function:

for any real number z > 0. Using the Sigma notation, this is also written as

This series can be derived from the above Taylor series. It converges more quickly than the Taylor series, especially if z is close to 1. For example, for z = 1.5, the first three terms of the second series approximate ln(1.5) with an error of about 3×10−6. The quick convergence for z close to 1 can be taken advantage of in the following way: given a low-accuracy approximation y ≈ ln(z) and putting

the logarithm of z is:

The better the initial approximation y is, the closer A is to 1, so its logarithm can be calculated efficiently. A can be calculated using the exponential series, which converges quickly provided y is not too large. Calculating the logarithm of larger z can be reduced to smaller values of z by writing z = a · 10b, so that ln(z) = ln(a) + b · ln(10).

A closely related method can be used to compute the logarithm of integers. From the above series, it follows that:

If the logarithm of a large integer n is known, then this series yields a fast converging series for log(n+1).