Log10n

Log10(n) denotes the base-10 logarithm of n. It is the exponent x such that 10^x = n and is defined for positive real numbers n. The base-10 logarithm is widely used in scientific, engineering, and numerical contexts because it maps multiplicative relationships into additive ones.

Notation and domain

Common notations include log10(n), log_10 n, or “log base 10 of n.” The domain is n > 0, and the range is all real numbers. The function is increasing on its domain, with log10(n) → -∞ as n → 0+, and log10(n) → ∞ as n → ∞. It is the inverse function of the exponential function 10^x.

Properties

Key identities include log10(1) = 0 and log10(10) = 1. The logarithm of a product equals the sum of logarithms: log10(ab) = log10(a) + log10(b). The logarithm of a power satisfies log10(a^k) = k log10(a). The change-of-base formula relates base-10 logarithms to other bases: log10(n) = ln(n) / ln(10), where ln is the natural logarithm.

Examples

log10(2) ≈ 0.3010, and log10(100) = 2. When n is a power of 10, log10(n) is an integer equal to the exponent.

Applications

Base-10 logarithms are used to determine orders of magnitude, express numbers in scientific notation, and in scales such as pH (which uses -log10[H+]). They also appear in various engineering contexts, including certain scales of sound and signal measurements.

Computational notes

In software, log10 is typically available as a standard function. If only natural logs are available, compute log10(n) as ln(n)/ln(10).