Logarithms, exponentials, and power laws appear frequently in signal analysis, and especially in hearing-motivated techniques. It is important to understand the reasons for their use, and to be able to recognize when they are inappropriate, and how to modify such mappings to make them more practical and robust.

Logarithms and Power Laws

Engineers like to describe signals and their spectra—and systems that process them—in logarithmic units. Our hearing is sometimes described as logarithmic, along both the loudness dimension and the pitch (or frequency) dimension. So we need to understand what this means, what's powerful and useful about logarithms, and what their limitations are as a conceptual model for perception of loudness and pitch in hearing.

As the Britannica (1797) cryptically explains, logarithms are “the indices of the ratios of numbers to one another; being a series of numbers in arithmetical progression, corresponding to others in geometrical progression; by means of which, arithmetical calculations can be made with much more ease and expedition than otherwise.” That is, logarithms were an invented way to make multiplication not much harder than addition, long before the logarithm was understood as a mathematical function. The logarithm function is also of great importance as the inverse of the exponential function, as we discuss in a later section.

A power law, on the other hand, is a remapping through a power, or exponentiation, such as a square, or a square root. Power laws also come in function/inverse pairs: the square and square root, or cube and cube root, or Nth power and Nth root (1/N power) in general, are such pairs.