log10V

Log10V denotes the base-10 logarithm of a quantity V, written as log10(V) or log10 V. It represents the exponent to which 10 must be raised to obtain V. In pure mathematics, V must be a positive real number; the real-valued function is undefined for V ≤ 0. Complex extensions exist but involve multi-valued logarithms.

For V > 0, log10(V) is continuous and strictly increasing. The derivative is 1/(V ln 10), and the function is concave down on its domain. It satisfies the standard logarithm rules, such as log10(ab) = log10(a) + log10(b) and log10(a^k) = k log10(a). The value at 1 is log10(1) = 0. The inverse function of log10 is the exponential base-10 function, 10^x. Change of base can express other bases: log_b(V) = log10(V) / log10(b).

In practical use, the argument V is often a ratio or a quantity normalized by a reference value, because the logarithm is only dimensionless when applied to a dimensionless quantity. Thus expressions like log10(V/Vref) are common in science and engineering. Log10V is widely used to linearize multiplicative data, compress large ranges, and facilitate comparison across orders of magnitude. In signal processing and acoustics, related scales use 20 log10(|V2/V1|) for amplitude ratios and 10 log10(P2/P1) for power ratios.

Numerical examples: log10(1) = 0, log10(10) = 1, log10(0.01) = -2. The function grows slowly: a tenfold increase in V raises log10(V) by only 1.