5* 
LOGARITHMS. 
897 
a r&x preyed by o, with decimals; from 10 to ioo, they 
are expreffed by i, with decimals; from ioo to 1000, they 
are expreiTed by s, with decimals ; from 1000 to xq,oooo, 
they are expreffed by 3, with decimals; &c. Thefe num¬ 
bers o, 1, 2, 3, Sec. are called indices, and the index is al¬ 
ways lefs by unity than the number of figures in the inte¬ 
gral part of the correfponding natural number. One me¬ 
thod of finding the logarithms of the intermediate numbers 
between 1 and 10, 10 and too, 100 and 1000, Sec. has been 
from the equation \o x —b, by finding x, having given b. 
The logarithm of .76348 154-8827977; for io 4 ' 8 8 2 7 977 
= 76348; divide by xo, and we get 10.3-8*2-7977—. 
divide again by 10, and we get ro 2 '* 8 2 7 977 = 763-48 ; 
and fo on. Hence, if any number be multiplied or di¬ 
vided by 10, the index of its logarithm will be increafed 
or diminilhed by 1, the decimal part remaining the fame. 
If therefore we continue to divide by 10 till the number 
becomes lefs than unity, and the index be negative, the 
decimal part is Hill politive ; on which account, the nega¬ 
tive fign is put over the index, becaufe that is the only 
figure affected by it. When therefore the negative fign is 
put over the index, it is to be urtderftood that the index is 
negative, and the decimal part pofitive ; but, when the ne¬ 
gative fign is put before the index, it is here to be under- 
itood, that both index and decimal part are negative. 
If the feries be continued downwards below 1, as io 5 , 
xo ! , io*, io°, 10“*, io -2 , io -3 , &c. the refpeclive loga¬ 
rithms are 3, 2, 1, o, —1, —2, —3, &c. Therefore the 
logarithm of any number which (lands between any two 
terms of the firft: feries, has for its index the index of the 
lead of thofe two terms, together with a decimal which is 
always pofitive. 
Thus the'logarithm of any number between 
•10 s 
io ! 
IO 1 
10° 
.10 1 
I0~ 5 
and 
and 
anti 
and 
and 
and 
io* 
10° 
10“ 
10“ 
io" 
is 2, -f- dec. 
is 1, -f- dec. 
is o, -j- dec. 
is T, -j- dec. 
is "2, -j- dec. 
Is + dec. &c. 
See. 
The logarithm thus expreffed is the true logarithm. Some¬ 
times, inftead of writing the index negative, 10 is added 
to it, and the index becomes pofitive ; in this cafe, the 
logarithm of any number between 
jo° and io -1 is 9, 4- dec. 
io~ t and io~ a is 8, dec. 
io -2 and io~ 3 is 7, -j- dec. See. Sec. 
But here, when we come to io~ i0 , the index becomes 
negative ; in this cafe therefore, to preferve the index po¬ 
fitive, it is ufual to add xoo to the index; hence, the lo¬ 
garithm of any number between 
io° and xo~< is 99,-1-dec. 
io~‘ and xo -2 is 98,-1-dec. 
so -2 and io~* is 97, + dec. See. Sec. 
In thefe cafes, the index is 10 or 100 greater than its true 
value, and in operations this mult be confidered. 
Hence, we have the following gradation of numbers 
and their logarithms, the logarithms of decimal numbers 
being written each way. 
Number. 
True Loga¬ 
rithms. 
Logarithm 
of Decimal 
Numbers, 
adding 10 
to the Index 
Logarithm of 
Decimal Num¬ 
bers, adding xop 
to the Index. 
76348 
7634-8 
763-48 
ij 76 - 34-8 
> 7^6348 
J T6348 
1 *076348 
g -0076348 
} -00076348 
4-8827977 
3-8827977 
2-8827977 
1-8827977 
0-8827977 
’» -8827977 
2 8827977 
T8827977 
T 8827977 
9 8827977 
8 8827977 
7 - 8 S 27977 
6.8827977 
99-8827977 
98-8827977 
97 8827977 
968827977 
The negative index denotes how many places the firft: 
Vgl. XII. No. 881. 
fignificant. figure of the decimal number is below unity; 
and, when 10, or 100, is added to the index, how much 
fuch index wants of 10, or 100, denotes the fame. The 
negative index, however, is that which ftands in the regu¬ 
lar fcale of logarithms, and always reprefenrs the true lo¬ 
garithm of a decimal number, and of that one number 
only ; whereas, the logarithm of a decimal expreffed by 
adding 10 or 100 to the index, is 10 or soo too great, and 
expreffes alfothe logarithm of a number greater than unity. 
Thus, 4-8827977 is the logarithm of 76348 ; and, conii- 
dering it as the logarithm of a decimal, the index of the 
logarithm being increafed by 10, it is a Kb, the logarithm 
of o 0000076348. By ufing the negative index, there is no 
danger of a miltake, and every fource of error fhould be 
cut off; we fhall therefore derive all our conclnfious in 
terms of the true logarithm. 
At the end of any operation where the logarithm of a 
decimal is taken by adding 10 to the index, and that lo¬ 
garithm is added, the 10 added muft be fubtradled ; and, if 
there be any number, r, of fuch logarithms added, then 
xor muft be fubtradted, that the refult may be the true lo¬ 
garithm. But, if a logarithm of this kind is to be jub- 
tradled, then io muft be added to the refulting index, to 
give the true logarithm ; for, by fubtrabling a quantity 
which is 10 above its true value, you make the refult 10 
lefs than its true value, and therefore 10 muft be added to 
give the true logarithm. In like manner, if you add xoo to 
the index, you muft, in the former ca fe, fubtracl 100 from 
the index for every fuch logarithm added-, and, in the lat¬ 
ter cafe, you muft add 100 for the logarithm which is fub- 
tradled-, and the refult will be th e true logarithm. 
When it is required to incorporate leveral logarithms 
by addition and fubtraflion, it will be more convenient to 
convert the fubtraflion into an addition, by writing down, 
inftead of the logarithm to be fubtrabted, what it wants 
of io-ooooooo, which you may very readily do, by wait¬ 
ing down what the iirft figure on the right wants of 10, 
and what every other figure wants of 9 ; this is called the 
arithmetical complement. For inftance, if the logarithm be 
7-4693875, its arithmetical complement is 2-5306:25. If 
one or more figures on the right be ciphers, write ciphers 
in their place, and take the firft fignificant figure on the 
right from 10, and the other figures from 9. Thus, if the 
logarithm be 5-3864000, its arithmetical complement is 
4-6136000. In any operation therefore, having taken the 
arithmetical complements of all the quantities to be fub- 
t rafted, the number of which we will fuppofe to be r, and 
added the whole together, you muftfubtraft xor from the 
index. The reafon of this is evident; for to add what a 
number wants of 10 muft evidently make a quantity 
greater by 10 than if you had lubtrafted that number. 
For inftance; 14 + 6 is greater by 10 than 14—4. lithe 
index of the quantity whofe arithmetical complemexst 
you want to take be greater than 10, you may write down 
what the index wants of 20, and, after the addition, fub¬ 
tract 20 from the index. 
When you have a logarithm with a negative index, it 
will fometimes be convenient to reduce the whole to a 
negative quantity; and this is done, by ftubtrafting 1 from 
the index, taking the arithmetical complement of the de¬ 
cimal part, and putting the fign —before the index, fo as 
to affeft the whole; for by this operation you increafe the 
value of the index by unity, and diminifh that of the de¬ 
cimal part by unity; and therefore the value of the loga¬ 
rithm is not altered. Thus, ’J-5972684 = — 2-40273x6 ; 
thefe we (hall call negative logarithms. This reduction will 
often be found convenient when you want to multiply ox- 
divide a logarithm with a negative index ; for otherwife, 
you muft, in general, multiply or divide the index, and the 
decimal part, feparately. And here, the refulting quan¬ 
tify muft be reduced back again to its negative index and 
pofitive decimal (by taking the arithmetical complement 
of the decimal part, adding 1 to the index, and putting 
the negative fign _over it), in order to ren u-r it conforma¬ 
ble to the icale oi ■ logarithms given in the common Ta- 
X 10 T bies. 
