Log10x

Log10x, often written as log10(x) or log10 x, denotes the common logarithm of x, the exponent to which 10 must be raised to obtain x. In other words, log10 x = y if and only if 10^y = x.

Definition and domain: The function is defined for x > 0, with log10 x yielding a real number y. As x increases from 0 to infinity, log10 x increases from negative infinity to positive infinity, making the function strictly increasing on its domain.

Key properties: log10(xy) = log10 x + log10 y; log10(x^k) = k log10 x; log10(1) = 0; log10(1/x) = -log10 x; log10(a/b) = log10 a − log10 b. These identities follow from the laws of exponents and aid in simplifying expressions and solving equations.

Change of base and computation: log10 x can be expressed as ln x / ln 10, where ln is the natural logarithm. This enables conversion between bases and is used when only natural logarithms are available.

Examples: log10 100 = 2, since 10^2 = 100; log10 0.01 = −2, since 10^−2 = 0.01.

Applications and notes: The common logarithm is widely used in scientific notation, data scaling and normalization, pH calculations, and scales such as decibels. It is undefined for x ≤ 0. In software and calculators, log often denotes base-10 logarithm, while ln denotes the natural logarithm; some environments use explicit base notation, e.g., log10(x).