-0CO 
lOGARITH M S. 
Ex. 2. What is theproduld fif 784 X 0-000079 X 00000036? 
.Here n — 13 5 hence, 
2-8945161 log. of - 784 
1-8976271 - - - - 79 
1-5563025 - - 36 
T-3+82457 
0-0000002229696 Product. 
Here the index of the fum is 6, from which fubtrafi 1377, 
•and the remainder is —7 ; therefore the firit fignificant 
figure is 7 places below unity. 
Thus the product of decimal faftors, as well as of whole 
numbers, may be always found by taking the logarithms 
of whole numbers only. 
Division by Logarithms. 
Rule.-—From the logarithm of the dividend fubtrafl 
the logarithm of -the divifor, and the remainder is the lo¬ 
garithm of the quotient. 
Ex. 1. Divide 2004-64 by 34. 
3-3020364 log. of - 2004-64 
1-5314789 - - - - 34 
17705S7S 
58-96 Quotient. 
Ex. 2. Divide 19 by 72. 
3-2787536 log. of 
i'85733*5 —-- 
19 
72 
1 -4214211 
- 0-2638889 Quotient. 
Here, as we carry 1 from the decimal to the index, we 
have to fubtrafl 2 in the index from 1 ; the remainder is 
therefore—1, which (hows that the firft fignificant figure 
of the quotient is 1 place below unity. In all cafes the 
operation is the fame, if the negative index be ufed for 
the logarithms of all quantities below unity. 
If the dividend be lefs, and the divifor be greater than 
-unity, and you write the logarithm of the former by add- 
in°- 10 to the index, you muft add 10 to the index of the 
logarithm of the latter before you fubtrafl. For, making 
k .each index equally too great, you get the true difference. 
Ex. Divide 0-000084 by 714. 
By adding 10 to the index : 
5-9242793 log. of - 0-000084 
£-8536982- - - -71+ 
7 -0705811 
0-000000117647 Quotient. 
T0705811 
0-000000117647 Quotient. 
5 ' 359 02I 9 
228571-4 Quotient. 
Here, 10 added to 1 make u, from which fubtract 6, and 
the remainder is 5. 
f By the negative index : 
1-9822712 log. of 96 
T 6232493 - - - 0-00042 
5-3590219 
228571-4 Quotient. 
Here,—4 fubtrafted from r, the remainder is 5. 
If the divifor and dividend be both lefs than unity, and 
their logarithms be written by adding 10 to the index, 
the fubtracfion gives the true refult. For, the true index 
of each logarithm being equally increal'ed, their difference 
is the true difference. 
Ex. Divide 0-2 by 0-00057. 
By adding 10 to the index : 
9-3010300 log. of - 0‘2 
6 - 755 8 749 -. - - - 0-00057 
a’ 54 - 5‘551 
350-877 Quotient. 
By the negative index -. 
T'3010300 log. of ... o"2 
"+'7558749- - - 0-000057 
2-5451551 
350-877 Quotient. 
Here, carrying 1 to — 4, the fum is — 3, which fubtrafted 
from — 1, the remainder is 2. The operation by the ne¬ 
gative index is preferable to that by the other method, as 
there is no variety of cafe6. 
If the divifor and dividend confifl of factors, of which 
fome, or all, are decimal numbers, the quotient may be 
found by the following Rule: Let the dividend be 
a X b X c X Sec. containing n decimals in all the factors 
together ; and let the divifor be iX<X“X &c. to r fac¬ 
tors, containing m decimals in all the factors ; and let 
A, B, C, See. and S, T, U, Sec. be the refpeftive values of 
a, b, c, See. and s, t , u. Sec. confidering their fignificant 
figures as whole numbers; then the logarithm of the quo¬ 
tient — log. A 4 - log. B 4- log. C 4- Sec. -j- ar. co. log. S, 
-}- ar. co. log. T 4- ar. co. log. U -J- Sec. -J- m — n — tor. 
„ . • . , . , A X B X C+&c. _ 
For, by the nature ot decimals, --- 
, „ , S X T X U X Sec. „ 
a X b X c X &c. and---= sX(X"X See, 
hence, io"~" X 
A X B x C X &c. fl X ^ X e X Sec. 
.Here, 10 added to 2 make 12, which taken from 5, the 
^remainder is —7. 
By the negative index : 
T9242793 log. of - - 0-000084 
2-8536982 - — * ■ 7 J 4 
If the dividend be greater, and the divifor lefs, than 
unity, and you write the logarithm of the latter by add¬ 
ing 10 to the index, you muft add 10 to the index of the 
logarithm of the former, before you fubtraft. This is 
true, for the reafon before given. 
Ex. Divide 96 by 0-00042. 
By adding 10 to the index : 
.1-9822712 log. of - - - 96 
6-6232493 -- ■ 0-00042 
S x T x U X Sec. ~ s x t X « X Sec. 
therefore the log. of the required quotient = log. A - 4 - log. 
B 4- log. C -f Sec. + ar. co. log. S 4- ar... co. log. T -j- 
ar. co. log. U 4 * Sec. + m — n — ior. 
,, rrrr . ■ ,r , r 8 4 X 0 00769 X 0-683 
Ex. What is the value of —----———-? 
598 X 0-0000146 x 0-039 
Here, n — 8, m = 10, m — n -= 2, r = 3, ror = 30, and 
m—n — io?-2= — 28; hence, 
1.9242793 log. of - - 84 
2-8859263 — —— - - - 769 
•2-8344207 -- - - - 683 
7- 2232988 ar. co. leg. of - 59 s 
7 8356471 —-- - - 146 
8- 4089354- - * 39 
3-1125076 
- 1295-71 Quotient. 
The index of the fum of the logarithms is 31, front 
which fubtrafl 28, and the remainder is 3 the index. By 
this method the operation is extremely plain and eafy, and 
you have to take the logarithms of whole numbers only. 
Involution by Logarithms. 
Rule.—Multiply the logarithm of the root by the index 
of the power, and the product is the logarithm of the 
* Ex. 
