LOG10

Log10, or the common logarithm, is the logarithm with base 10. For a positive real number x, log10(x) is the exponent to which 10 must be raised to produce x; that is, 10^y = x if and only if y = log10(x). The function is defined only for x > 0, and its values span the real numbers; as x approaches 0 from the right, log10(x) tends to -infinity, and as x grows, log10(x) tends to infinity. In notation, log10(x) is sometimes written as log10 x or log(x) when the base is understood to be 10. The change-of-base formula log10(x) = ln(x)/ln(10) relates it to the natural logarithm.

Key properties include: log10(ab) = log10(a) + log10(b); log10(a^k) = k log10(a); log10(1) = 0. These follow from the general laws of logarithms and underpin many algebraic manipulations.

Historically, the base-10 logarithm was developed in the early 17th century. John Napier introduced logarithms, and Henry Briggs later promoted the base-10, or common, logarithm system. Common logarithms are widely used in science and engineering to simplify multiplication, division, powers, and roots; they also underlie scales such as the Richter scale, decibels, and pH calculations. Log10 scales compress large ranges of values, so equal distances on a log10 plot correspond to orders of magnitude. In programming and numerical work, log10 is a standard function; it is monotone increasing on x > 0, with 10^x as its inverse.