PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: High Entry average rating: Very high
logarithm (Definition)

Definition.

Three real numbers $x, y, p$ , with $x,y>0$ and $x \neq 1$ , 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.
$\displaystyle \log_x(yz)$ $\displaystyle = \log_x(y) + \log_x(z)$    
$\displaystyle \log_x \left( \frac{y}{z} \right)$ $\displaystyle = \log_x(y) - \log_x(z)$    
$\displaystyle \log_x(y^z)$ $\displaystyle = z \log_x(y)$    
$\displaystyle \log_x(1)$ $\displaystyle = 0$    
$\displaystyle \log_x(x)$ $\displaystyle = 1$    
$\displaystyle \log_x(y) \log_y(x)$ $\displaystyle = 1$    
$\displaystyle \log_y(z)$ $\displaystyle = \frac{\log_x(z)}{\log_x(y)}$    

By the very first identity, any logarithm restricted 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 $e = 2.71828\ldots$ . 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 universal. 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 [ Biography]




Anyone with an account can edit this entry. Please help improve it!

"logarithm" is owned by rmilson. [ full author list (4) ]
(view preamble | get metadata)

View style:

See Also: entropy, complex logarithm

Also defines:  base, natural logarithm, ln, log

Attachments:
approximation of the log function (Derivation) by Mathprof
Briggsian logarithms (Topic) by pahio
rigorous definition of the logarithm (Derivation) by rspuzio
table of natural logarithms (Data Structure) by PrimeFan
examples of logarithms simplifying calculations (Example) by PrimeFan
logarithmic scale (Definition) by pahio
Log in to rate this entry.
(view current ratings)

Cross-references: theory, information, function, Eulerian number, applications, computers, calculators, properties, addition, multiplication, additive function, integers, positive, identities, algebraic, number, operation, property of the exponential, monotonicity, exponential, relation, real numbers
There are 78 references to this entry.

This is version 18 of logarithm, born on 2002-02-21, modified 2007-12-27.
Object id is 2367, canonical name is Logarithm.
Accessed 45314 times total.

Classification:
AMS MSC26-00 (Real functions :: General reference works )
 26A09 (Real functions :: Functions of one variable :: Elementary functions)
 26A06 (Real functions :: Functions of one variable :: One-variable calculus)

Pending Errata and Addenda
None.
[ View all 5 ]
Discussion
Style: Expand: Order:
forum policy
Values of the logarithm base by kfgauss70 on 2007-11-17 20:25:08
Some authors state that the case x=1 is to be excluded from the logarithm function domain.
[ reply | up ]

Interact
post | correct | update request | add derivation | add example | add (any)