log10b

Log10b, written as log10(b) or log base 10 of b, is the base-10 logarithm of a number. It represents the exponent to which 10 must be raised to obtain b. In mathematical notation, for positive b, log10(b) = ln(b) / ln(10), where ln denotes the natural logarithm. The use of natural logarithms in the fraction provides a convenient base-change formula that applies to any positive argument.

Domain and range: In real-valued contexts, log10(b) is defined only for b > 0. The range is all real numbers, with larger positive inputs yielding larger results and approach toward negative infinity as b approaches zero from the positive side. For negative inputs or zero, the real-valued logarithm is undefined, though complex extensions exist via logarithms with multiple values.

Key properties: log10(1) = 0 and log10(10) = 1. The function satisfies log10(xy) = log10(x) + log10(y) and log10(x^k) = k log10(x) for positive x. It is strictly increasing on its domain, and the derivative is d/dx log10(x) = 1 / (x ln(10)) for x > 0. These rules support algebraic manipulation and simplification in equations involving powers and products.

Applications: log10 is widely used in science and engineering to express orders of magnitude and to transform data that span large ranges. Examples include the pH scale, which uses pH = -log10[H+], and the decibel scale, which involves 10 log10 of intensity ratios. It also underpins many algorithms and data analysis techniques that require multiplicative data to be analyzed additively.

Notation: The term common logarithm is sometimes used to refer to log10, reflecting historical use. In programming and calculators, log10 is a standard function, with input restrictions aligning with the real-domain definition.