Log10sumk

Log10sumk is a function that returns the base-10 logarithm of the sum of k nonnegative real numbers. Given numbers a1, a2, ..., ak with ai ≥ 0, log10sumk(a1,...,ak) = log10( sum_{i=1}^k ai ). If the sum is zero, the result is undefined in the real numbers (often represented as −∞ in the extended real line). If negative inputs are allowed, the expression is not defined without additional handling.

Computation and numerical stability are common considerations. Directly computing the sum then applying log10 can suffer from overflow when ai are large or underflow when many small terms are present. A numerically stable approach uses a logarithmic transformation. If bi = log10(ai), then log10(sum ai) = M + log10( sum_{i=1}^k 10^{bi−M} ), where M = max_i bi. This avoids producing extremely large intermediate values. If inputs are given in natural logarithms, convert to base 10 via log10(ai) = ln(ai)/ln(10). Alternatively, if the inputs are already in log-space, the same max-and-shift trick applies with natural logs and a base-10 conversion at the end.

Domain and variants: log10sumk is typically defined for nonnegative inputs. With negative inputs, one would need a different formulation or restrict to magnitudes. In programming, log10sumk may be implemented directly or as a variant of the log-sum-exp technique, often used in statistics and machine learning to maintain numerical precision in log-domain computations.

Example: for a = [1, 10, 100], sum = 111, so log10sumk(a) = log10(111) ≈ 2.0458.