"how numbers are stored in computers"

The logarithmic mean function

This theorem analyzes the logarithmic mean function , which appears in floating-point error analysis and logarithmic approximations. This function has useful and well-behaved properties in the range , which the theorem formalizes.

Specifically, it proves that for all , and the derivative is bounded in magnitude with . The proof uses the Taylor expansion of , which is an alternating series:

Since the series alternates with decreasing terms, the sum lies below the first term, hence . Furthermore, because all terms past the first are negative for , we can estimate , and when , this means . So overall, for small positive .

The derivative, , also expands into an alternating Taylor series. Like , it is well-behaved and bounded in absolute value by , because the leading term in 's expansion is negative and therefore subsequent terms decay quickly. This gives a bound of at most for small values of .

These bounds make and its derivative particularly stable and reliable in numerical computations — especially in error estimation for logarithmic expressions and in the analysis of floating-point division and logarithmic mean approximations.

Theorem

If , then for , and the derivative satisfies .

Proof

Observe that the Taylor expansion of is:

This is an alternating series with terms that decrease in magnitude. For , this implies , since . Also, because the series alternates starting from 1 and subtracts terms, it must be less than or equal to 1. Therefore:

Now consider the derivative . Its Taylor series is also alternating and takes the form:

For , this is again an alternating series with decreasing terms, and so we can bound the derivative:

Since , this implies , and the lower bound is , so .