LOG 



LOG 



Find the number correfponding to the logarithm 

 2.5450987. 

 Next greater log. 2.54319^^6 Given log. 2.5430987 

 Next lefa log. 2.^^^oj.i2 Next lefs z. 5430742 



"I'abular differ. 



1244 



DlfTcr 



245 



i244)24J-oo{i9 

 1244 



Therefore ^549. 219 is the number fought, the firft four 

 figures being the number anfvvering to the leafl logarithm. 



To ptrform arithmetical operation by logarithms. 



Multiplication by logarithms. — Takeout the logarithms of 

 the factors from the table, and their .fum 'will be the loga- 

 rithm of the produft fought ; then, by means of the table, find 

 the natural number anfivering to that logarithm, which will 

 be the product required. Obferving to add what is carried 

 from the decimal part of the logarithm to the affirmative 

 index, or indices, or fubtraft it from the negative. Alfo 

 adding the indices together if they are of the fame kind ; 

 fre. all pofitive, or all negative, but to fubtraft them if they 

 be of different kinds, prefixing the fign of the greater to 

 the remainder. Thus, 



Multiply together .7684, 68.42, and .34S76 

 log. of .7684 = — 1.8855874 

 log. of 68.42 = 1.8351831 

 log. of .34876 = — 1.5425267 



Produft 18.3357 = 1.2632972 



Divi/ion by logarithms.— Yiere the logarithms are to be 

 taken out as above, and then the logai'ithm of the divifor 

 mull be Aibtrafted from that of the dividend, and the re- 

 mainder will be the logarithm of the quotient fouglit, ob- 

 ferving to change the fign of the index of the divifor from 

 affirmative to negative, or from negative to affirmative ; then 

 take the fum of the indices, if they be of the fame kind, or 

 fubtraft them if they be of different kinds, prefixing the 

 fign of t!ie greater for the index. Alio, if i is borrowed in 

 the left-hand place of the decimal part of the logarithm, 

 add it to the index of tlie divifor when that index is affir- 

 mative, but fubtraft it when negative ; then let the fign be 

 changed, and worked with as before. Thus, for eiample, 



Du-ide 37.149 by 523.67 



log- .S7-I49 = i-5^'9947i 



log. 523.67 = 2.7190577 

 Quotient .0709397 = — 2.8508894 



•' Involution, or raifmg of powers by logarithms. — Multiply 

 the logarithm of the given number ijy the index of the power 

 to which it is to be raifed ; and the produft will be the 

 logarithm of the power required. But in multiplying a 

 logarithm with a negative index, the producl will be nega- 

 tive, but what is carried from the decimal part will be pofi- 

 tive, and mull, therefore, in that cal'e, be fubtraCted from 

 that prodiift. 

 Vol. XXI. 



Hence, to find the cube of .30714^ 

 log. of .307146 = - 1.4873449 



i 



Power .0289758 — 2.4620347 



Evolution, or the extra^ion of roots by logarithms. — Divide 

 the logarithm of the given number by the index of the 

 power, the root of which is to be extra£led, and the quotient 

 will be the logarithm of the root required ; obferving, that 

 if the index of the logarithm be negative, as many units muft 

 be borrowed as will make it cxaftly divifiblc, and fo mav.y 

 units mull then be carried to the decimal part of the loga- 

 rithm, and the divilion carried on as ufual. 



Required the cube root of .12345 



l"g- 12345 3)- 1-09149" 



Root 



497925 



1. 697 1 637 



Thefe are the moll fimple cafes in which logarithms are 

 introduced into arithmetical operations ; the application of 

 them to more complex cafes, as in Tri>;onometry, Menfura- 

 tion, &c. will be explained under the refpeftive heads. 



LoGAniTii.M, Imaginary, h ufedfor the logarithm of nega- 

 tive and imaginary quantities, fuch as — a, ^' — a, &c. 

 Thus, alfo, the fluents of certain imaginary fluxionary ex- 



preffions, fuch as , —7 , &c, arc imagi- 



^ .1- y — I 2i.v y- I ^ 



nary logarithms. Euler Analyf. Infin. vol. i. p. 72. 74. 



X 



The expreffion — reprefents the fluxion of the logarithm 



X 



of X, and the fluent, therefore, of — is the logarithm of x ; 



but no logarithm can reprefent the fluent of - 



-, which 



X ^' ~ I 

 is therefore called an imaginary logarithm. 



However, when thefe imaginary logarithms occur in the 

 folutions of problems, they may be transformed into cir- 

 cular arcs or feCtors ; that is, the imaginary logarithm, or 

 •imaginary hyperbolic feftor becomes a real circular feclor. 

 See Bernouilii, Oper. torn. i. p. 400. and p. 512. Mac- 

 laurin's Fluxions, art. 762, feq. Walmefly, Anal. de« 

 Mef. p. 63. 



LOGE, in Geography, a town of Germany, in the 

 county of Hoya ; 20 miles S.Vv'. of Nienburg. 



LOGGERHEAD Key, or El Contoy, a fmall ifland 

 in the bay of Honduras, near the coaft of Yucatan. N. 

 lat. 21" 25'. W. long. 87' 45'. 



LOGGERHEAT, in the Sea Language, denotes a 

 large round ball of iron, with a long handle for heating- 

 pitch. 



LOG-HOUSES, houfesin America, which are generally 

 the firfl that are ereSed on any new fettlement, and which are 

 cheaper than any others in a country where wood abounds. 

 The fides conhll of trees jull fquared, and placed hori- 

 zontally one upon the other ; the ends of the logs of one 

 fide relling alternately on the ends of thofe of tlie adjoining 

 fides, in notches ; the interilices between the logs are Uopped 

 with clay ; and the roof is covered with boards or fhingles, 

 which arc fnuill pieces of wood in the fliape of fiates or 

 tiles, &:c. which are ufed for that purpofe, with few excep- 

 tions, throughout America., Thefe ii.ibitations are not very 

 fio-htlv, but wiien well built they are warm and comfortable, 

 and laii for a bug time, ijome of tiiea» are built with brick 

 O o " sr 



