log 
II. a. Constructed of I. ,.,'s; consisting of logs: log 4 (log), n. [Heb. %/<.] A Hebrew liquid 
as, a /(/ cabin ; a ><l l'"i't or bridge Log cabin, measure, the seventy-second part of a bath, or 
, , 
a cabin or hut built of lot's, unhewn or hewn, notched near 
the ends and laid one upon another, and having the inter 
Btlces filled wild 
mud or plaster. 
lag cabins are of- 
ten used as dwell- 
ings in poor 
thinly set 
or 
ettled re 
gions where tiin- 
her abounds. 
g-cabin cam- 
. in V. S. 
. the electoral 
canvass for the 
presidency in 1S4U, 
in which represen- 
tations of log ca- 
bins anil barrels nf 
hard cider were 
carried in the pro- 
cessions of the par- 
tizans of William Henry Harrison. One of his opponents. 
wrongly attributing to him a humble origin, had spoken of 
him as one who had lived in a log cabin and drunk hard 
cider, and the expression was caught up by his adherents 
and made a party cry. Log-cabin quilt, a patchwork 
quilt of a particular design. [U.S.] 
Reluctantly she slipped her hook under the log-cabin 
quilt, and said "Come in." Harper's Mag., LXXVI. 36. 
Log canoe, a canoe hollowed out of a single log. Log 
house, a house built of logs fitted together, and smoothed 
on the inside, or on both sides. Log houses in new or 
about a pint. It seems to have been of i',;il.y- 
loniau origin, being one sixtieth of a maxis. 
He shall take . . . three tenth deals of Hue flour for a 
meat offering, mingled with oil, and one log of oil. 
Lev. xiv. 10. 
log. Tho abbreviation of logarithm. Thug, 
I<HJ. 'A = 0.4771213 is an equation giving the 
Mil in of the logarithm of 3. 
logan, n. See liiggan. 
Loganiaceae (16-ga-ni-a'se-e), n. pi. [NL. (End- 
licher, 1836), ^ Logania, the typical genus, -f 
-aceo;.] An order of gamopetalous exogens, 
characterized by opposite,usually entire leaves, 
with stipules which adhere to the leaf-stalks 
or are combined in the form of interpetiolary 
sheaths. The flowers usually grow In terminal or axil- 
lary cymes, and are four- or live-parted, with an Inferior 
calyx, the stamens inserted on the corolla-tube, and a fruit 
which is capsular, drupaceous, or a berry. The order in- 
cludes SOgenera and about 350 species, either herbs, shrulm, 
or trees., which are dispersed throughout tropical and sub- 
tropical regions. The plants are Mtter and highly poi- 
sonous ; the poison-nut, Strychnos nttx-wmica, belongs to 
this order, and several other species are used in medicine. 
Besides Logania, an Australian genus and type of the or- 
der, it includes Gelnemium, the yellow jessamine of the 
southern United States, and Spiyelia, the pinkroot or 
worm-grass. 
thickly wooded regions of North America are often of logaoedic (log-a-e'dik). a. and . [< LL. loga- 
considerable size and well finished. oedicus, < LGr. fayaoiotnot;, logaccdic, < Gr. /<5- 
log 1 (log), v. ; pret. and pp. logged, ppr. log- j, ofi speech, prose (see Logoa), + dotor/, song: 
ging. [< log'-, .] I.t trann. To cut into gee ode.] I. a. Literally, prose-poetic ; in one. 
logs. )>ros., noting a variety of trochaic or iambic 
When a Tree Is so thick that after it is loy'd it remains verse in which dactyls are combined with tro- 
stlll too great a Burthen for one Man, we blow It up with chees or anapests with iambi : so called be- 
1 .";: and Keel. 
Gunpowder. Dumpier, Voyages, II. ii. 80. 
II. intrant. To cut down trees and get out 
logs from the forest for sawing into boards, 
etc. : as, to engage in logging. 
log' 2 (log), [= D- G- ' </> *- Sw. logg = Dan. 
lull, a ship's log, a piece of wood that 'lies' in 
the water; diff. from Icel. lag, a felled tree (> 
E. log 1 ), but from the same ult. source, namely 
Icel. liggja = Sw. ligga = Dan. ligge, etc., lie: 
see Mel.] j. Naut., an apparatus for measuring 
the rapidity of a ship's motion. 
The most common form consists of a 
log-chip, or thin quadrant of wood, of 
about five inches radius, fastened to 
a line wound on a reel. When the log- 
chip is thrown overboard, its motion 
Udeadened on striking the water, and 
its distance from the ship, measured 
after a certain time on the line (which 
Is allowed to run out), gives approx- 
imately the speed of the ship. The 
chip is loaded with lead on the arc 
side to make it float upright. At 12 
or 15 fathoms from the chip a white 
rag marks off the stray-line, a quan- 
tity sufficient to let the log-chip get 
clear of the vessel before time is 
marked. The rest of the line, which is from 160 to 200 
fathoms long, Is divided into equal parts by bits of 
string stuck through the strands and distinguished by the 
number of knots made in each, or in some similar way, as 
by colored rags ; hence these divisions are called knots. 
The length of a knot must bear the same proportion to the 
length of a nautical mile (see mite) that the time during 
which the line is allowed to run out bears to one hour. 
Thus, using a twenty-eight second glass, 28 : 3600 : : 47.3 
feet (the usual length of a knot) : 6080 feet (the usually 
received length of a sea-mile). Many other devices have 
been Invented to perform the functions of the log, which 
generally include a brass fly or rotator connected with 
mechanism acting as an index. In some cases the whole 
machine is towed astern of the ship, and must be hauled 
in to be examined ; with the taffrau-log, the register is fas- 
tened to the tatfrall and the fly Is towed astern. 
Hence 2. The record of a ship's progress, 
or a tabulated summary of the performance of 
tho engines and boilers, etc. ; a log-book. 
Electric log, an apparatus devised for measuring the 
speed of water-currents, or the speed and distance trav- 
eled by ships at sea, with the aid of electricity. With the 
second kind mentioned under electric, the distance run is 
indicated by a pointer on a dial, which shows the number 
of turns made by a screw towed behind the vessel. Elec- 
trical conductors are incased in the tow-line, and the cir- 
cuit is closed at intervals of a stated number of turns, 
thus operating an indicator on deck. Electric logs have 
not come into practical use. Ground-log, a form of log 
adapted for showing the direction and speed of passage of 
a vessel over the ground in shoal water. It consists of an 
ordinary log-line, with a hand lead of 7 or 9 pounds substi- 
tuted for the log-chip. When used, the lead remains fixed 
at the bottom, and the line shows the path and speed of 
the ship and the effect of any current which may exist. 
Rough log, in the rmted States navy, the original manu- 
script of a snip's log. To heave the log. See heave. 
log" (log), r. t.; pret. and pp. logged, ppr. log- 
giiig. R log". "] 1. To record or enter in 
the log-book. 2. To exhibit by the indication 
of the log, as a rate of speed by the hour: as, 
the ship loijx ten knots. 
log s t (log), r. /. [The apptir. orijr. of the freq. 
form loggi'i-X, q. v. ('f. also li>ggn>i.] To move 
to and fro; rock. SIT /<></</hi</-rock. 
N'iltural 
numbers. 
0.1 
1 
10 
100 
1000 
10000 
100000 
Napier's 
logarithms. 
OMB 
100000000 
76974149 
1101018 
301)22447 
Successive 
differences. 
23025851 
23025861 
-16129K6 
HOS6851 
nmea 
U01B8H 
It will thus be seen that if four numbers. A, B, C, D, are in 
proportion, so that A :8 = C : D, then theirfonrlogarithms 
satisfy the equation, log A log B = log C log D ; so 
that, to work the rule of three with logarithms, we sim- 
ply substitute for each number its logarithm and pro- 
ceed as usual, only that in every case we perform addi- 
tion instead of multiplication and subtraction instead of 
division ; and the result is the logarithm of the answer. 
(6) As now understood, a system of loga- 
rithms, besides the two essential characters 
set forth above, has a third, namely that the 
logarithm of 1 is (I. ibis 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 bate of the system, must 
chees or anapests with iambi: so called be- 
cause this apparent irregularity seems to ap- 
proach the non-observance of metrical laws 
characteristic of prose. These dactyls and anapests 
are not, however, full dactyls or anapests of four times or 
morse, but cyclic dactyls or anapests of only three times, 
equivalent therefore In measure to trochees or iambi. A 
single long syllable is also used in some places in several 
forms of logaoadic verse to represent a complete foot. 
This long is equal not to two but to three shorts, and is 
therefore equivalent to a trochee. Irrational longs that 
Is, longs reduced to the value of a short also occur in 
the theses. A basis sometimes precedes the series. Re- 
cent metricians use the epithet logaoedic of mixed meters 
(see mixed) in general. Ancient writers classed many 
logaoedic meters as Ionic, epionic, cboriambic, epicho- 
riambic, or antispastlc. Among the more familiar loga- 
oedic meters are the Olyconic, Pherecratic, Asclepiadic, 
Sapphic, and Alcaic. See basis, 9, and cyclic, 3. 
II. n. A verse of the character denned above, 
logarithm (log'a-rithm or -riTHm), n. [Cf. F. 
logarithme = Sp". logaritmo = Pg. logarithmo = 
It. logaritmo = D. G. logarithme = Dan. loga- 
ritme = Sw. logaritm (< E.); < NL. logarithmtitt 
(NGr. ).oyaptO/tof), < Gr. Aojof, proportion, ratio 
(see Logos), + apiff/i6c., a number: see arithme- 
tic.] (a) An artificial number, or number used 
in computation, belonging to a series (or sys- 
tem of logarithms) having the following prop- 
erties : First, every natural or ordinary number, integral 
or fractional, has a logarithm in each system of loga- 
rithms ; and conversely, every logarithm belongs to a nat- 
ural number, called its antil<njarithm. 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 num 
her 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 preced- 
ing 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 sub- 
tracting the approximate value of one from that of the 
other, is in error by 1 in the last decimal place. 
logarithm 
be raised in order to produce the natural number, or an- 
tilogarithm. Thu (base)i<* = *. At the time loga- 
rithms were Invented fractional exponents had not been 
tl ubt of, ami even decimals, as we conceive them, were 
little us. 'I, the decimal point not having yet appear. <l; 
consequently, the last definition of tlic localithm, wlii.-b 
is now the usual one, was not at first possible. With log- 
aiitlnns in the modern sense, the rule for solving pro- 
portions still holds, but Is secondary to the following fun- 
damental 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 deter- 
mine the circumference of the earth In Inches, knowing 
that Its radius is 3H5h miles. We take out from a table of 
logarithms the logarithms of all the numbers which have 
to be multiplied together, as follows : 
Names of quantities. r!u*!br. logarithms. 
Radius of the earth in miles 3958 3.5974858 
Ratio of diameter to radius 2 0.3010800 
Ratio of circumference to diameter 3. 1415927 (1.4971499 
One mile In feet 5280 3.7226389 
One foot in Inches 12 1.0791312 
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 1575S91000. 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 fol- 
low as corollaries. Thus, to divide one number by an- 
other, subtract the logailthm 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 loga- 
rithm 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 UK; to the thousandth power. The common 
logarithm of 1.08 is 0.0253068653; 1,000 times this is 
25.3058653, which is the logarithm of 2022384 followed by 
19 ciphers, or say 20 quadrillions 223S40 trillions, In the 
English numeration. To give an idea of the advantage of 
logarithms in trigonometrical calculations, it may be men- 
tioned that to find the altitude of the sun from its hour- 
angle and declination with logarithms requires seven num- 
bers to be taken out of the tables and two additions to be 
performed, while the solution of the eame problem with 
a table of natural sines requires, as before, the taking out 
of seven numbers from the tables, and besides eight ad- 
ditions and two halvings. There are two systems of loga- 
rithms in common use, the hyperbolic, natiirat,oi Napierian 
or Nfpierian (not Napier's own) logarithms in analysis, 
and common, decimal, or Brigysian logarithms In ordinary 
computations. The base of the system of hyperbolic loga- 
rithms is 2.71S281828459. This kind of logarithm derives 
its name from its measuring the area between the equi- 
lateral hyperbola, an ordinate, and the axes of coordinates 
when these arc the asymptotes; but the chief character- 
istic of the system is that, x being any number less than 
unity, 
log (1 + x) = x - J z2 + J x3 } z4 4 etc. 
Thus, the hyperbolic logarithm of 1.1 Is calculated as fol- 
X 0.100000000 1x2 0.005000000 
lx'-l 0.000333333 f*4 0.000025000 
!z5 0.000002000 Jz6 0.000000167 
: i~ 0.000000014 lx 0.000000001 
0.100335347 0.005025168 
0.005026168 
log 1.1 0.096310179 
By the skilful application of this principle, with some 
others of subsidiary importance, the whole table of natu- 
ral logarithms has been calculated. The logarithms of 
any other system, in the modem sense, are simply the pro- 
ducts of the hyperbolic logarithms into a factor constant 
for that system, called the modulus of the system of loga- 
rithms ; 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 loga 
rlthm consists of an integer part and a decimal : the for- 
mer is called the index or characteristic, the latter the 
mantissa. The characteristic depends only upon the po- 
sition of the decimal point, and not at all upon the suc- 
cession of significant figures ; the mantissa depends en- 
tirely upon the succession of figures, and not at all upon 
the position of the decimal point. Thus, 
log 12345 
log 1234.5 
log 123.45 
4.09141)11 
3.0914911 
2.IHM 11)1] 
The characteristic of a logarithm is equal to the number 
of places between the decimal point and the first signifi- 
cant 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 j, we may write 1.69S9700. The sec- 
ond and most usual way is to augment the logarithm 
by 10 or by 100, thus forming a logarithm in the ori- 
ginal sense of the word. Thus, 0. SO 10300 would be 
written !).69h9700, the characteristic in this case being 
9 less the number of places between the decimal point 
and the first significant figure. I.oRarithms were In- 
vented and a table published in 1614 by John Napier of 
Scotland ; but tbe kind now chiefly In use were proposed 
by his contemporary Henry Briggs. professor of geometry 
In Rresham College in London. The first extended table 
