logarithm

Definition.

Three real numbers $x,y,p$, with $x,y>0$ and $x\ne 1$, are said to obey the logarithmic relation

$${\log }_x(y)=p$$

if they obey the corresponding exponential relation:

$$x^p=y.$$

Note that by the monotonicity and continuity property of the exponential operation, for given $x$ and $y$ there exists a unique $p$ satisfying the above relation. We are therefore able to says that $p$ is the logarithm of $y$ relative to the base $x$.

Properties.

There are a number of basic algebraic identities involving logarithms.

${\log }_x(yz)$ $={\log }_x(y)+{\log }_x(z)$

${\log }_x\left({\displaystyle \frac{y}{z}}\right)$ $={\log }_x(y)-{\log }_x(z)$

${\log }_x(y^z)$ $=z{\log }_x(y)$

${\log }_x(1)$ $=0$

${\log }_x(x)$ $=1$

${\log }_x(y){\log }_y(x)$ $=1$

${\log }_y(z)$ $={\displaystyle \frac{{\log }_x(z)}{{\log }_x(y)}}$

By the very first identity, any logarithm restricted (http://planetmath.org/RestrictionOfAFunction) to the set of positive integers is an additive function.

Notes. In essence, logarithms convert multiplication to addition, and exponentiation to multiplication. Historically, these properties of the logarithm made it a useful tool for doing numerical calculations. Before the advent of electronic calculators and computers, tables of logarithms and the logarithmic slide rule were essential computational aids.

Scientific applications predominantly make use of logarithms whose base is the Eulerian number $e=2.71828\mathrm{\dots }$. Such logarithms are called natural logarithms and are commonly denoted by the symbol $\ln$, e.g.

$$\ln (e)=1.$$

Natural logarithms naturally give rise to the natural logarithm function.

A frequent convention, seen in elementary mathematics texts and on calculators, is that logarithms that do not give a base explicitly are assumed to be base $10$, e.g.

$$\log (100)=2.$$

This is far from . In Rudin’s “Real and Complex analysis”, for example, we see a baseless $\log$ used to refer to the natural logarithm. By contrast, computer science and information theory texts often assume 2 as the default logarithm base. This is motivated by the fact that ${\log }_2(N)$ is the approximate number of bits required to encode $N$ different messages.

The invention of logarithms is commonly credited to John Napier [ http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Napier.htmlBiography]

Title	logarithm
Canonical name	Logarithm
Date of creation	2013-03-22 12:25:21
Last modified on	2013-03-22 12:25:21
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	21
Author	rmilson (146)
Entry type	Definition
Classification	msc 26A09
Classification	msc 26A06
Classification	msc 26-00
Related topic	Entropy
Related topic	ComplexLogarithm
Defines	base
Defines	natural logarithm
Defines	ln
Defines	log