Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
 noun The power to which a base, such as 10, must be raised to produce a given number. If n^{x} = a, the logarithm of a, with n as the base, is x; symbolically, log_{n} a = x. For example, 10^{3} = 1,000; therefore, log_{10} 1,000 = 3. The kinds most often used are the common logarithm (base 10), the natural logarithm (base e), and the binary logarithm (base 2).
from The Century Dictionary.
 noun An artificial number, or number used in computation, belonging to a series (or system of logarithms) having the following properties:
 noun As now understood, a system of logarithms, besides the two essential characters set forth above, has a third, namely that the logarithm of 1 is 0.
 noun The sum of these logarithms is 9.1974808, which we find by the table to be the logarithm of a number comprised between 1575690000 and 1575091000. To obtain a closer approximation, we should have to carry the logarithms to more places of decimals; but this would be useless, since the radius of the earth is only given to the nearest mile. From this fundamental rule several subsidiary rules follow as corollaries. Thus, to divide one number by another, subtract the logarithm of the divisor from that of the dividend, and the antilogarithm of the remainder is the quotient; to take the reciprocal of a number, change the sign of the logarithm, and the antilogarithm of the result is the reciprocal; to raise a number to any power, multiply the logarithm of the base by the exponent of the power, and the antilogarithm of the product is the power sought; to extract any root of a number, divide the logarithm of that number by the index of the root, and the antilogarithm of the quotient is the root sought. For example, what is the amount of $1 at interest at 6 per cent. compounding yearly for 1,000 years? We must here raise 1.06 to the thousandth power. The common logarithm of 1.06 is 0.0253058653; 1,000 times this is 25.3058653, which is the logarithm of 2022384 followed by 19 ciphers, or say 20 quadrillions 223840 trillions, in the English numeration. To give an idea of the advantage of logarithms in trigonometrical calculations, it may be mentioned that to find the altitude of the sun from its hourangle and declination with logarithms requires seven numbers to be taken out of the tables and two additions to be performed, while the solution of the same problem with a table of natural sines requires, as before, the taking out of seven numbers from the tables, and besides eight additions and two halvings. There are two systems of logarithms in common use, the hyperbolic, natural, or Napicrian or Neperian (not Napier's own) logarithms in analysis, and common, decimal, or Briggsian logarithms in ordinary computations. The base of the system of hyperbolic logarithms is 2.718281828459. This kind of logarithm derives its name from its measuring the area between the equilateral hyperbola, an ordinate, and the axes of coordinates when these are the asymptotes; but the chief characteristic of the system is that, x being any number less than unity, Thus, the hyperbolic logarithm of 1.1 is calculated as follows:
 noun By the skilful application of this principle, with some others of subsidiary importance, the whole table of natural logarithms has been calculated. The logarithms of any other system, in the modern sense, are simply the products of the hyperbolic logarithms into a factor constant for that system, called the modulus of the system of logarithms; and each system in the old sense is derivable from a system in the modern sense by adding a constant to every logarithm. The base of the common system of logarithms is 10, and its modulus is 0.4342944819. A common logarithm consists of an integer part and a decimal: the former is called the index or characteristic, the latter the mantissa. The characteristic depends only upon the position of the decimal point, and not at all upon the succession of significant figures; the mantissa depends entirely upon the succession of figures, and not at all upon the position of the decimal point. Thus
 noun The characteristic of a logarithm is equal to the number of places between the decimal point and the first significant figure. Logarithms of numbers less than unity are negative; but, negative numbers not being convenient in computation, such logarithms are usually written in one or other of two ways, as follows: The first and perhaps the best way is to make the mantissa positive and take the characteristic only as negative, increasing, for this purpose, its absolute value by 1, and writing the minus sign over it. Thus, in place of writing –0.3010300, which is the logarithm of ½, we may write 1.6989700. The second and most usual way is to augment the logarithm by 10 or by 100, thus forming a logarithm in the original sense of the word. Thus, –0.3010300 would be written 9.6989700, the characteristic in this case being 9 less the number of places between the decimal point and the first significant figure. Logarithms were invented and a table published in 1614 by John Napier of Scotland; but the kind now chiefly in use were proposed by his contemporary Henry Briggs, professor of geometry in Gresham College in London. The first extended table of common logarithms, by Adrian Vlacq, 1628, has been the basis of every one since published. Abbreviated l. or log.
from the GNU version of the Collaborative International Dictionary of English.
 noun (Math.) One of a class of auxiliary numbers, devised by John Napier, of Merchiston, Scotland (15501617), to abridge arithmetical calculations, by the use of addition and subtraction in place of multiplication and division.
 noun the difference between a logarithm and the number ten.
 noun See under
Binary .  noun logarithms of which the base is 10;  so called from Henry Briggs, who invented them.
 noun tables of logarithms constructed for facilitating the operation of finding the logarithm of the sum of difference of two quantities from the logarithms of the quantities, one entry of those tables and two additions or subtractions answering the purpose of three entries of the common tables and one addition or subtraction. They were suggested by the celebrated German mathematician Karl Friedrich Gauss (died in 1855), and are of great service in many astronomical computations.
 noun a logarithm (devised by John Speidell, 1619) of which the base is e (2.718281828459045...);  so called from Napier, the inventor of logarithms.
 noun See under
Logistic .
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun mathematics For a number , the
power to which a givenbase number must be raised in order to obtain . Written . For example, because and because .
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 noun the exponent required to produce a given number
Etymologies
from The American Heritage® Dictionary of the English Language, 4th Edition
from Wiktionary, Creative Commons Attribution/ShareAlike License
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Examples

Someone mentioned the word logarithm to me the other day, a word I had not heard for several decades.

Per ShangKeng Ma, an entropy can be defined as the logarithm of the phasespace volume explored by the system over a given timescale; the states relevant for thermodynamics (mumble mumble microcanonical mumble mumble) would be the decategorification of the states used at the statisticalmechanical level.
Arrow of Time FAQ Sean 2007

An exponential is a function that increases or decreases by a certain fixed multiplicative factor whenever you change the argument by an additive shift while the logarithm is the inverse function.

A logarithm is a name for a power to which you raise a given number, called the base, to get the number whose logarithm it is.
The Past Through Tomorrow Heinlein, Robert A. 1967

Imagine thinking that a logarithm is a soothing thing instead of the exciting thing that is truly is.
Wired Campus 2010

Slide 6: Elliptic Curve Cryptography (ECC) Solution is ECC Its security comes from the elliptic curve logarithm, which is the DLP (Discrete Logarithm Problem) in a group defined by points on an elliptic curve over a finite field.

You know, Adam, somehow I think I'd trust your assessment a bit more if you could spell the word "logarithm".
RealClimate 2009

Standard of living, as measured by the natural logarithm of gross domestic product per capita at purchasing power parity.

“Find the logarithm of the ninth root of three,” Mrs. James ordered.
Times Squared Julia DeVillers 2011

All he needed was the value of the natural logarithm of 2.
SuperCooperators Martin A. Nowak 2011
moby commented on the word logarithm
The Inverse of the Power Function as Division is the inverse of Multiplication.
December 29, 2006