LOGARITHM LONG, JOHN DAVIS
LOGAU, FRIEDRICH LONG BRANCH
LOGIA LONGCLOTH
LOGIC LONG EATON
LOGOCYCLIC CURVE, STROPHOID LONGEVITY
LOGOGRAPHI LONGFELLOW, HENRY WADSWORTH
LOGOS LONG FIVES
LOGOTHETE LONGFORD (county of Ireland)
LOGROÑO (province of Spain) LONGFORD (town of Ireland)
LOGROÑO (Spanish town) LONGHI, PIETRO
LOGROSCINO, NICOLA LONGINUS, CASSIUS
LOGWOOD LONG ISLAND
LOHARU LONG ISLAND CITY
LÖHE, JOHANN KONRAD WILHELM LONGITUDE
LOHENGRIN LONGLEY, CHARLES THOMAS
LOIN LONGMANS
LOIRE (river of France) LONGOMONTANUS, CHRISTIAN SEVERIN
LOIRE (department of France) LONGSTREET, JAMES
LOIRE-INFÉRIEURE LONGTON
LOIRET LONGUEVILLE
LOIR-ET-CHER LONGUEVILLE, ANNE GENEVIÈVE
LOISY, ALFRED FIRMIN LONGUS
LOJA LONGWY
LOKEREN LÖNNROT, ELIAS
LOKOJA LONSDALE, EARLS OF
LOLLARDS LONSDALE, WILLIAM
LOLLIUS, MARCUS LONS-LE-SAUNIER
LOLOS LOO
LOMBARD LEAGUE LOOE
LOMBARDO LOOM (water-birds)
LOMBARDS LOOM (weaving machine)
LOMBARDY LOÓN
LOMBOK LOOP
LOMBROSO, CESARE LOOSESTRIFE
LOMÉNIE, ÉTIENNE CHARLES DE LOOT
LOMOND, LOCH LOPES, FERNÃO
LOMONÓSOV, MIKHAIL VASILIEVICH LOPEZ, CARLOS ANTONIO
LOMZA (government of Russian) LOPEZ DE GÓMARA, FRANCISCO
LOMZA (town of Russia) LOP-NOR
LONAULI LOQUAT
LONDON (Canada) LORAIN
LONDON (capital of England) LORALAI
LONDON CLAY LORCA
LONDONDERRY, EARLS OF LORCH (Prussian town)
LONDONDERRY, STEWART (VANE) LORCH (town of kingdom of Württemberg)
LONDONDERRY, ROBERT STEWART LORD, JOHN
LONDONDERRY (county of Ireland) LORD
LONDONDERRY (town of Ireland) LORD ADVOCATE
LONG, GEORGE
LOGARITHM (from Gr. [Greek: logos], word, ratio, and [Greek: arithmos],
number), in mathematics, a word invented by John Napier to denote a
particular class of function discovered by him, and which may be defined
as follows: if a, x, m are any three quantities satisfying the equation
a^x = m, then a is called the base, and x is said to be the logarithm of
m to the base a. This relation between x, a, m, may be expressed also by
the equation x = log(a) m.
_Properties._--The principal properties of logarithms are given by the
equations
log(a) (mn) = log(a)m + log(a)n, log(a)(m/n) = log(a)m - log(a)n,
log(a)m^(r) = r log(a)m, log(a)[root r]m = (1/r)log(a)m,
which may be readily deduced from the definition of a logarithm. It
follows from these equations that the logarithm of the product of any
number of quantities is equal to the sum of the logarithms of the
quantities, that the logarithm of the quotient of two quantities is
equal to the logarithm of the numerator diminished by the logarithm of
the denominator, that the logarithm of the rth power of a quantity is
equal to r times the logarithm of the quantity, and that the logarithm
of the rth root of a quantity is equal to (1/r)th of the logarithm of
the quantity.
Logarithms were originally invented for the sake of abbreviating
arithmetical calculations, as by their means the operations of
multiplication and division may be replaced by those of addition and
subtraction, and the operations of raising to powers and extraction of
roots by those of multiplication and division. For the purpose of thus
simplifying the operations of arithmetic, the base is taken to be 10,
and use is made of tables of logarithms in which the values of x, the
logarithm, corresponding to values of m, the number, are tabulated. The
logarithm is also a function of frequent occurrence in analysis, being
regarded as a known and recognized function like sin x or tan x; but in
mathematical investigations the base generally employed is not 10, but a
certain quantity usually denoted by the letter e, of value 2.71828
18284....
Thus in arithmetical calculations if the base is not expressed it is
understood to be 10, so that log m denotes log10 m; but in analytical
formulae it is understood to be e.
The logarithms to base 10 of the first twelve numbers to 7 places of
decimals are
log 1 = 0.0000000 log 5 = 0.6989700 log 9 = 0.9542425
log 2 = 0.3010300 log 6 = 0.7781513 log 10 = 1.0000000
log 3 = 0.4771213 log 7 = 0.8450980 log 11 = 1.0413927
log 4 = 0.6020600 log 8 = 0.9030900 log 12 = 1.0791812