# logarithm

## Definitions

• WordNet 3.6
• n logarithm the exponent required to produce a given number
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Webster's Revised Unabridged Dictionary
• n Logarithm (Math) One of a class of auxiliary numbers, devised by John Napier, of Merchiston, Scotland (1550-1617), to abridge arithmetical calculations, by the use of addition and subtraction in place of multiplication and division.The relation of logarithms to common numbers is that of numbers in an arithmetical series to corresponding numbers in a geometrical series, so that sums and differences of the former indicate respectively products and quotients of the latter; thus,0 1 2 3 4 Indices or logarithms1 10 100 1000 10,000 Numbers in geometrical progression
Hence, the logarithm of any given number is the exponent of a power to which another given invariable number, called the base, must be raised in order to produce that given number. Thus, let 10 be the base, then 2 is the logarithm of 100, because 102 = 100, and 3 is the logarithm of 1,000, because 103 = 1,000.
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Century Dictionary and Cyclopedia
• n logarithm An artificial number, or number used in computation, belonging to a series (or system of logarithms) having the following properties: First, every natural or positive number, integral or fractional, has a logarithm in each system of logarithms; and conversely, every logarithm belongs to a natural number, called its antilogarithm. Second, in each system of logarithms, the logarithms corresponding to any geometrical progression of natural numbers are in arithmetical progression: that is, if each natural number of the series is obtained from the preceding one by multiplying a constant factor into this preceding one, then each logarithm may be obtained from the preceding one by adding a constant increment or subtracting a constant decrement. This is shown, for the system of Napier's logarithms, in the following table. It must be said that logarithms are, in general, irrational numbers, and their values can only be expressed approximately, being carried to some finite number of decimal places. Owing to the neglected places, it will often happen that the difference between two logarithms, obtained by subtracting the approximate value of one from that of the other, is in error by 1 in the last decimal place.
• n logarithm 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. This being admitted, a simpler definition can be given of the logarithm, viz.: a logarithm is the exponent of the power to which a number constant for each system, and called the base of the system, must be raised in order to produce the natural number, or antilogarithm. Thus (base)log x = x. At the time logarithms were invented fractional exponents had not been thought of, and even decimals, as we conceive them, were little used, the decimal point not having yet appeared; consequently, the last definition of the logarithm, which is now the usual one, was not at first possible. With logarithms in the modern sense, the rule for solving proportions still holds, but is secondary to the following fundamental rule: The sum of the logarithms of several numbers is the logarithm of the continued product of those numbers. For example, let it be required to determine the circumference of the earth in inches, knowing that its radius is 3958 miles. We take out from a table of logarithms the logarithms of all the numbers which have to be multiplied together, as follows:
• n logarithm 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 hour-angle 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:
• n logarithm 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
• n logarithm 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.
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Chambers's Twentieth Century Dictionary
• n Logarithm log′a-rithm (of a number) the power to which another given number must be raised in order that it may equal the former number: one of a series of numbers having a certain relation to the series of natural numbers by means of which many arithmetical operations are simplified
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## Etymology

Webster's Revised Unabridged Dictionary
Gr. lo`gos word, account, proportion + 'ariqmo`s number: cf. F. logarithme,
Chambers's Twentieth Century Dictionary
Gr. logos, ratio, arithmos, number.

## Usage

### In literature:

Invention of logarithms by Lord Napier, England.
"The Great Events by Famous Historians, Volume 11" by Various
The complement of the logarithm of a sine, tangent, or secant.
"The Sailor's Word-Book" by William Henry Smyth
He had learned to use logarithms.
"Great Men and Famous Women. Vol. 4 of 8" by Various
They brought her a treatise on logarithms by the Rev.
"Stories of Authors, British and American" by Edwin Watts Chubb
In after life I, of course, used logarithms for the higher branches of science.
"Personal Recollections, from Early Life to Old Age, of Mary Somerville" by Mary Somerville
He used logarithms, and proved the accuracy of his work by different methods.
"From Farm House to the White House" by William M. Thayer
Instead of going into logarithms, Henry went into shorthand.
"A Great Man" by Arnold Bennett
On leaving school he took up mathematics as a specialty and invented a system of logarithms based on the number 12 instead of 10.
"Elementary Theosophy" by L. W. Rogers
In 1594, he made a contract with Napier of Merchistoun, the inventor of Logarithms.
"James VI and the Gowrie Mystery" by Andrew Lang
While he is going over his logarithms to know what should be done, the commonest seaman on board could set all to rights.
"Sporting Scenes amongst the Kaffirs of South Africa" by Alfred W. Drayson
The above definitions of logarithms, &c., relate to cases in which n and p are whole numbers, and are generalized later.
"Encyclopaedia Britannica, 11th Edition, Volume 2, Slice 5" by Various
And do you understand logarithms?
"Wilson's Tales of the Borders and of Scotland" by Various
If a^x = N, x is said to be the logarithm of N to the base a.
"The New Gresham Encyclopedia" by Various
These formulae give, in the case of k = 0.1a, values certain to eight logarithmic decimal places.
"Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6" by Various
After reading the laudatory sentences bestowed upon the inventor of logarithms, it is very amusing to find J.
"Sir Thomas Urquhart of Cromartie, Knight" by John Willcock
But as to eloquence, he knows no more about it than a table of logarithms.
"Sketches of Reforms and Reformers, of Great Britain and Ireland" by Henry B. Stanton
He left school at sixteen, after having mastered geometry and trigonometry, and having learned to use logarithms.
"Historic Fredericksburg" by John T. Goolrick
He is strong only in his knowledge of Greek verbs and logarithms.
"At Start and Finish" by William Lindsey
In trigonometry the Professor and I had a disagreement touching a little matter of logarithms.
That is, the stream lines will be logarithmic spirals.
"Encyclopaedia Britannica, 11th Edition, Volume 14, Slice 1" by Various
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### In news:

This post offers reasons for using logarithmic scales, also called log scales, on charts and graphs.
When Should I Use Logarithmic Scales in My Charts and Graphs.
There are two main reasons to use logarithmic scales in charts and graphs.
A comparison of linear and logarithmic ( log ) scales.
Does the Brain Work Logarithmically .
This illustrates the logarithmic spiral curve r=exp(0.17*theta) under continuous clockwise rotation / expansion.
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### In science:

Note also that the logarithmic terms in both (18) and (19) yield the sums of geometric progression with arguments of logarithms related by eq.(17).
On the Fourier transformation of Renormalization Invariant Coupling
The logarithm of the average susceptibility of the largest system (L = 64) is plotted verses the logarithm of T − Tc , for temperatures above Tc and for different values of Tc .
Full reduction of large finite random Ising systems by RSRG
In Fig. 4(b), we have used a linear ﬁt for the logarithm of the maximum of −∂χ/∂T plotted versus the logarithm of L, from which we obtain the value of (γ + 1)/ν .
Full reduction of large finite random Ising systems by RSRG
In Fig. 5 we use a linear ﬁt for the logarithm of Tc (L) − Tc plotted versus the logarithm of L.
Full reduction of large finite random Ising systems by RSRG
In (b), the logarithm of the maximum of −∂χ/∂T is plotted verses the logarithm of L.
Full reduction of large finite random Ising systems by RSRG
FIG. 6: The logarithm of the susceptibility, scaled by a factor of 1/L2−η , is plotted versus the logarithm of T − Tc , scaled by a factor of L−1/ν .
Full reduction of large finite random Ising systems by RSRG
General logarithmic solutions Here we consider hypergeometric equations which have logarithmic points, but do not have terminating hypergeometric solutions.
Degenerate Gauss hypergeometric functions
The base-2 logarithm of x is denoted by log x, and the natural logarithm is denoted by ln x.
Capacity and Random-Coding Exponents for Channel Coding with Side Information
Behavior in the other tail. behavior of fM1 (θ) and its logarithmic derivatives as θ → ∞, and the behavior of fM2 (θ) and its logarithmic derivatives as θ → −∞.
Saddlepoint approximation for moment generating functions of truncated random variables
It seems that we have reached a certain numerical limit related to the logarithmic contribution and the calculation of the non-logarithmic terms will be much more complicated than anything else done before.
Precision physics of simple atoms: QED tests, nuclear structure and fundamental constants
The other possibility for this logarithm to appear is a logarithmic integration over coordinate or momentum space which leads to ln(mchri/¯h), where hri is the characteristic size of the atomic state.
Precision physics of simple atoms: QED tests, nuclear structure and fundamental constants
We give sharp uniform bounds for exponentials of logarithmic potentials if the logarithmic capacity of the subset, where they are at most 1, is known.
Constructive Function Theory on Sets of the Complex Plane through Potential Theory and Geometric Function Theory
Figure 2 shows the maxima of the logarithm derivatives as a function of the logarithm of the lattice size L for p = 1 and p = 2.
Critical behavior of the spin-3/2 Blume-Capel model on a random two-dimensional lattice
FIG. 3: Plot of the logarithm of the modulus of the magnetization at the inﬂection point as a function of the logarithm of L = N 1/2 .
Critical behavior of the spin-3/2 Blume-Capel model on a random two-dimensional lattice
In Figs. 9 and 10 we plot the average logarithmic coupling and the density of free sites vs. the logarithmic ﬂow parameter, Γ, for the two distributions types (further described in the ﬁgure captions).
Transverse Meissner Physics of Planar Superconductors with Columnar Pins
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