logarithm base 10 — from formula anatomyThe base-10 logarithm, written as log10(x) or simply log(x), is the power to which 10 must be raised to obtain the number x. For example, log10(100) = 2 because 10^2 = 100. It is a fundamental tool in science for dealing with quantities that span many orders of magnitude, such as sound intensity (decibels), earthquake energy (Richter scale), or acidity (pH). Because it converts multiplicative relationships into additive ones, it simplifies analysis and visualization of exponential growth or decay.
The concept of common logarithms was introduced by John Napier in the early 17th century, but the base-10 version was developed by Henry Briggs in 1617, who collaborated with Napier to create tables of logarithms for practical calculation.
The logarithm is the inverse function of exponentiation with base 10. For any positive number x, log10(x) = y such that 10^y = x. It has the properties log10(a*b) = log10(a) + log10(b) and log10(a^b) = b * log10(a). These properties make it easy to manipulate large or small numbers and to linearize exponential data for regression analysis.
It is used in every field that deals with orders of magnitude, including acoustics (decibels), seismology (Richter scale), chemistry (pH), and information theory (bits and entropy). It also underpins logarithmic scales in graphing and data transformation.
The slide rule, a mechanical analog computer used for over 300 years, works by exploiting the property that logarithms turn multiplication into addition.
| M | variable | stellar mass — mass kilogram |
| L | variable | stellar luminosity — luminosity watt |
| M_sun | variable | solar mass — mass kilogram |
| L_sun | variable | solar luminosity — luminosity watt |
| a | constant | empirical slope of the relation — empirical slope = 3.5 |
| b | constant | empirical intercept of the relation — empirical intercept = 0 |
| log10 | operator | base-10 logarithm — logarithm base 10 |
| [X/Y] | variable | logarithmic abundance ratio relative to solar — abundance ratio |
| n_X | variable | number density of element X — number density per cubic metre |
| n_Y | variable | number density of element Y — number density per cubic metre |
| (n_X/n_Y)_⊙ | variable | solar ratio of number densities — number density ratio |
| log10 | operator | base-10 logarithm — logarithm base 10 |
| m | variable | apparent magnitude — apparent magnitude |
| M | variable | absolute magnitude — absolute magnitude |
| d | variable | distance to the object — distance parsec |
| log10 | operator | base-10 logarithm — logarithm base 10 |
| m | variable | apparent magnitude — apparent magnitude |
| H | variable | absolute magnitude — absolute magnitude |
| d | variable | distance to the object — distance parsec |
| \text{phase correcti | variable | correction for phase angle (e.g., for planets or asteroids) — phase correction |
| \log_{10} | operator | logarithm to base 10 — logarithm base 10 |
| E(B-V) | variable | Color excess between B and V bands — color excess |
| k(λ) | variable | Extinction curve at wavelength λ — extinction curve |
| F_λ | variable | Flux at wavelength λ — flux watt per metre squared |
| λ | variable | Wavelength — wavelength metre |
| log10 | operator | Base-10 logarithm — logarithm base 10 |
| \Delta[\mathrm{X}/\m | variable | Difference in logarithmic abundance ratio — metallicity difference |
| N_{\mathrm{X}} | variable | Number density of element X — number density per cubic metre |
| N_{\mathrm{Y}} | variable | Number density of element Y — number density per cubic metre |
| \log_{10} | operator | Base-10 logarithm — logarithm base 10 |
| \Delta | operator | Difference between two states — difference |
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