Logarithmically

Logarithmically is an adverb used to describe relationships, scales, or distributions governed by logarithmic functions or multiplicative processes. In mathematics, a quantity that grows logarithmically increases in proportion to the logarithm of its variable; for large n, f(n) ~ c log_b(n), where b is the logarithm base. Saying that a quantity increases logarithmically conveys slower growth than any power function.

In data visualization and measurement, "logarithmically" often describes a scale or spacing: a logarithmic scale uses successive values that increase by a constant factor, so equal distances correspond to multiplicative changes. Data plotted on a log scale or with logarithmically spaced samples can reveal multiplicative relationships or accommodate wide ranges.

In computer science, algorithms with logarithmic time complexity are described as running in logarithmic time, often written O(log n). Data structures like binary search trees yield logarithmic height, and several divide-and-conquer methods exhibit logarithmic growth in resource use.

In statistics and information theory, distributions may be described as logarithmically distributed when values are more evenly spread on a log scale, or when probabilities are proportional to 1/x, i.e., log-uniform in a specified interval.

Note that the base of the logarithm is usually determined by context (base 10, base e, or base 2); the behavior described by "logarithmically" is effectively the same up to a constant factor across bases.