1log10x

1log10x is not a standard mathematical notation. The expression most likely refers to the common logarithm of x, which is the logarithm to base 10 and is typically written as log10(x) or log_10 x. This article describes that function and its common properties.

Definition and basic properties. Let f(x) = log10(x). By definition, f(x) is the exponent to which 10 must be raised to produce x. The domain is x > 0, and the range is the set of all real numbers. The inverse function of f is the exponential function g(y) = 10^y, since log10(10^y) = y and 10^{log10(x)} = x. An equivalent expression is log10(x) = ln(x) / ln(10).

Key identities and features. For any positive x and any real a, log10(x^a) = a log10(x). Also log10(1/x) = -log10(x), and log10(10) = 1. The function is strictly increasing on its domain, with derivative f'(x) = 1 / (x ln 10). As x approaches 0 from the right, log10(x) → -∞; as x → ∞, log10(x) → ∞. The point x = 1 yields log10(1) = 0.

Applications and usage. Base-10 logarithms are widely used in science and engineering to handle data that span multiple orders of magnitude, such as quantities on the pH scale, decibels, and Richter-type measures. They also appear in data analysis, entropy calculations, and various numerical methods. Historically, the base-10 logarithm facilitated calculations before electronic calculators.

Notation note. If you encounter the concatenated form 1log10x, it is not standard and should be interpreted as a misprint or nonstandard notation. The accepted form is log10(x) or log_10 x.