log10a

Log10A denotes the logarithm of A with base 10, also written log10(A) or log_10 A. It answers the question: to what power must 10 be raised to obtain A? Specifically, log10 A = x means 10^x = A.

The function is defined for A > 0 and yields any real number as A ranges over positive real numbers. It is strictly increasing on its domain.

Key properties include: log10(1) = 0 and log10(10) = 1; log10(AB) = log10 A + log10 B; log10(A^k) = k log10 A; and log10(A/B) = log10 A − log10 B. Change of base allows converting to other bases: log10 A = ln A / ln 10. It is the inverse of the power function 10^x, so if y = log10 A, then A = 10^y.

Examples: log10 1000 = 3; log10 0.01 = −2; log10 7 ≈ 0.8451.

Applications and context: the base-10 logarithm, or common logarithm, is widely used in science and engineering. It underpins pH measurements (pH = −log10[H+]), and is involved in decibel calculations (power ratios use 10 log10, while amplitude ratios use 20 log10). Logarithmic scales compress large ranges of values and aid in data visualization, multiplicative processes, and order-of-magnitude estimation.

Notes: log10 is not defined for zero or negative real inputs in standard real-valued arithmetic. For matrices, log10 can be applied elementwise to a matrix with positive entries, but more general matrix logarithms require distinct definitions.