logarithm

Definition.

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

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

if they obey the corresponding exponential relation:

$$x^p=y.$$

Note that by the monotonicity and continuity property of theexponential 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 toaddition, and exponentiation to multiplication. Historically, theseproperties of the logarithm made it a useful tool for doing numericalcalculations. Before the advent of electronic calculators andcomputers, tables of logarithms and the logarithmic slide rule wereessential computational aids.

Scientific applications predominantly make use oflogarithms whose base is the Eulerian number $e=2.71828\mathrm{\dots }$.Such logarithms are called natural logarithms and are commonlydenoted 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 oncalculators, is that logarithms that do not give a base explicitly areassumed to be base $10$, e.g.

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

This is far from . In Rudin’s “Real andComplex analysis”, for example, we see a baseless $\log$ used torefer to the natural logarithm. By contrast, computer science andinformation theory texts often assume 2 as the default logarithm base.This is motivated by the fact that ${\log }_2(N)$ is the approximatenumber 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