10log10

10log10 is the expression ten times the base-10 logarithm of a positive quantity x, commonly written as 10 log10(x). The function is defined for x > 0 and is monotone increasing. It maps (0, ∞) to (-∞, ∞). Its derivative is 10 /(x ln 10). The inverse function is x = 10^(y/10). Since log10 x = ln x / ln 10, 10 log10 x can be computed from natural logs as 10 * (ln x / ln 10).

In decibels, the quantity is often a ratio P/P0, with dB = 10 log10(P/P0). For amplitude or field quantities, the factor 20 log10 is used because power is proportional to amplitude squared. Examples: P/P0 = 100 → dB = 20; P/P0 = 0.01 → dB = -20.

Notes: The decibel is a relative, dimensionless measure derived from a logarithmic scale. The 10 log10 form compresses wide-ranging ratios into a more manageable range and is widely used in fields such as acoustics, electronics, and communications. The choice of 10 versus 20 depends on whether the quantity being expressed is a power ratio or an amplitude/field ratio.

See also: base-10 logarithm, decibel, logarithm, power ratio, amplitude ratio.