labours in 1781, and their collection of paraphrases is still in use. Eleven of them are the composition of Logan, and others were revised or altered by him. In the same year he published his poems in a volume which attracted so much attention that a second edition was issued in 1782. It also included the "Ode to the Cuckoo," with which Edmund Burke was so pleased that when in Edinburgh he sought out Logan and complimented him as the author of the finest ode in the English language.
In 1783 he published a tragedy called Runnamede, which met with little success. In 1786 he resigned his charge at South Leith, retaining part of his stipend. He then went to London, where he devoted himself to literature. He was engaged on the management of the English Review, and in 1788 published a pamphlet on the charges against Warren Hastings. He died in December 1788.
A work on ancient history, published that year under the name of Dr Rutherford, rector of an academy at Uxbridge, is believed to have been the lectures written by Logan. His sermons were published in 1790-91, in two volumes, and have been several times reprinted. His poems were collected and published in 1812, with a memoir understood to be by the Rev. R. Douglas of Galashiels.
About forty years after Logan’s death what may be called the Bruce-Logan controversy arose by the publication in 1836 of a life of Michael Bruce, prefixed to an edition of his poems by the Rev. Dr MacKelvie. In this work there is claimed for Bruce the author ship of sixteen of the pieces in the volume issued by Logan in 1770. Logan was at the same time charged with having retained some of Brace’s MSS. entrusted to him, which housed in the revision of the paraphrases. These statements have been reiterated with much abuse of Logan in a memoir of Bruce prefixed to an edition of his works published in 1865, by the Rev. Dr Grosart. In this edition the paraphrases written by Logan are inserted as having been written by Bruce. The evidence, however, brought against Logan in these biographies of Bruce, being nearly altogether of a hearsay character, is not of much value, and it may be urged that Logan was not blamed during his life for any such literary delinquencies. If anything can be brought against him with justice, it is his publishing as his own, with very few alterations, the second Paraphrase, which is the composition of Dr Doddridge.
Within the last few years the various pieces in the volume of Bruce’s poems issued by Logan in 1770 have been subjected to careful criticism, and the statements made from personal knowledge by the Rev. Dr Robertson of Dalmeny, the college friend of Bruce and Logan, who was often referred to on the subject, must be held to be substantially correct. These will be found in a brochure by Dr David Laing, Ode to the Cuckoo (Edinburgh 1770), with remarks on its authorship, in a letter to J. C. Shairp, LL.D. , 1873. See also a paper by J. Small in the Brit, and For. Evan. Rev., 1877, and especially two papers by the Rev. R. Small, ibid., 1878.
LOGANSPORT, capital of Cass county, Indiana, U.S., is situated at the confluence of the Wabash and the Eel rivers, and on the Wabash and Erie canal, 75 miles north-west of Indianapolis. It is an important railway junction, and the trading-centre of an extensive agricultural district—dealing in grain, pork, and timber (poplar and black walnut). The Pittsburg, Cincinnati, and St Louis railroad maintains at this point large carriage-works, occupying 25 acres, and employing 600 men. The population was 11,198 in 1880.
LOGARITHMS. The definition of a logarithm is as follows:—if a, x, m are any three quantities satisfying the equation ax = 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 = loga m.
Properties. The principal properties of logarithms are given by the equations
| , | , |
| , | , |
![{\displaystyle \scriptstyle \log a\ {\sqrt[{r}]{m}}={\frac {1}{r}}\log a\ m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43a9b95c050b4cfacde23adc54443c7b30d3358f)
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 1rth 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 equal to 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 formulæ 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 |
The meaning of these results is that
| 1 = 10, | 2 = 100.3010300, | 3 = 100.4771213, | … |
| 10 = 101. | 11 = 101.0413927 | 12 = 101.0791812. |
The integral part of a logarithm is called the index or characteristic, and the fractional part the mantissa. When the base is 10, the logarithms of all numbers in which the digits are the same, no matter where the decimal point may be, have the same mantissa; thus, for example,
log 2.5613 = 0.4084604, log 25.613 = 1.4084604,
log 2561300 = 6.4084604, &c.
In the case of fractional numbers (i.e., numbers in which the integral part is 0) the mantissa is still kept positive, so that, for example,
log .25613 = 1.4084604, log .0025613 = 3.4084634, &e.
,
the minus sign being usually written over the characteristic, and not before it, to indicate that the characteristic only and not the whole expression is negative; thus
1.4084604 stands for − 1 + .4084604.
The fact that when the base is 10 the mantissa of the logarithm is independent of the position of the decimal point in the number affords the chief reason for the choice of 10 as base. The explanation of this property of the base 10 is evident, for a change in the position of the decimal points amounts to multiplication or division by some power of 10, and this corresponds to the addition or subtraction of some integer in the case of the logarithm, the mantissa therefore remaining intact. It should be mentioned that in most tables of trigonometrical functions, the number 10 is added to all the logarithms in the table in order to avoid the use of negative characteristics, so that the characteristic 9 denotes in reality 1, 8 denotes 2, 10 denotes 0, &c. Logarithms thus increased are frequently referred to for the sake of distinction as tabular logarithms, so that the tabular logarithm = the true logarithm + 10.
In tables of logarithms of numbers to base 10 the mantissa only is in general tabulated, as the characteristic of the logarithm of a number can always be written down at sight, the rule being that, if the number is greater than
