L O G A R 
Ex. To find, the logarithm of 
Here, by the rule, the 
logarithm is io + log. 3 — log. 5= 10*4771213— 0-6989700 
— 9-778^1513. Or the required logarithm is log. 3 — 
log. 5 = 0-4771113 — o-6989700 = T , 77Si5 1 3 the true lo¬ 
garithm. 
A Logarithm being given, to find the Number. 
To find the number to five figures anfwering to any given loga¬ 
rithm. —Rule. Neglect the index, and feek for the deci¬ 
mal part of the logarithm, the three firft: figures in the 
fecond column, and the other four in that or fome of the 
other columns; and even with thefe four figures in the 
firlt column, N, you have the firlt four figures of the re¬ 
quired number; and the fifth figure is that which Itands 
at the head of the column where the four figures were 
found; and the number of integral figures is greater by 
unity than the index. 
Ex. Let the given logarithm be a-4967775. Having found 
the three firlt decimal figures 496 in the fecond column, 
againft 7775 you find 3138 in the firft column, and at the 
bead of 7775 you find 9, which annexed to 3138 make 
313895 and the index being 2, the required number is 
313-89. 
If the logarithm be found exaftly, and the index be 
greater than 4, ciphers muft be added to the right of the 
five figures, till the whole number of figures be greater by 
unity than the index ; for, by increafing the index by 1, 
you make the correfponding number ten times greater. 
To find the number to fix or /even figures anfwering to any 
given logarithm. —Rule. The given logarithm, A, not 
being found exaftly in the Table, take the next lefts, B ; 
and in the column Pts, having at its head the difference 
between the logarithms next greater and lefts than the 
given logarithm, look for A — B, and againft it you have 
the fixth figure; but if A — B be not found exadftly, take 
the figure in the column Pts correfponding to the next lefts 
number than A — B, for the fixth figure required; then 
take the difference between A — B and the next lefts num¬ 
ber in the column Pts, and prefix a cipher to the right; 
and againft that number, or the nearelt to it, you have 
the feventh figure required ; and the number of integral 
figures is greater by unity than the index. 
Ex. What is the number correfponding to the logarithm 
6*4970385 ? The difference between the logarithms next 
greater and lefts than the given logarithm, is 139; this 
number therefore ftands at the head of the column Pts to 
be entered. 
Given log. - 6-4970385 = ,4 
Next lefts log. - 6 4970264=: B 
121 — A — B 
8 is the fixth figure - - - - 111 next lefts 
7 is the feventh figure - - - 100 
Now the firft five figures of the number anfwering to B are 
31407 ; therefore the required number is 3140787. 
Or the additional figures, as many as may be required, 
may be found thus : Add as many ciphers to the right 
of A — B as you want additional figures, and divide that 
quantity by the difference between the logarithm B and 
the next greater logarithm, and the quotient gives the 
figures required. Thus, in the lalt example, if we add 
two ciphers to A — B, it becomes 12100, which divided 
by 139 (the difference of the logarithm B and the next 
greater logarithm), the quotient is 87, the two next figures. 
If the given logarithm be the logarithm of a decimal 
with a potitive index, find the number correfponding to 
the decimal part, and prefix as many decimal ciphers to 
the left of it as the index wants of 9. 
Ex. Let the logarithm be 6-6380897 ; and let it. reprefent 
the logarithm of a decimal with a pofiiive index, or of a whole 
number with feven figures. Look into the Table, and againft 
I T H M S. 89^ 
the log. 6380897 you find the number 4346; and, the in¬ 
dex 6 being 3 lefts than 9, there muft be three ciphers 
added to the left 5 hence, the required decimal number is 
0-0004346. 
If the logarithm be written with a negative index, as 
T6380897, find the fignificant figures of the correfpond¬ 
ing number as before, and the index ftiows how many 
places the firft fignificant figure is below unity. 
Having thus far explained the principles of logarithms, 
we proceed to exemplify their ufe in numerical calculations. 
Multiplication by Logarithms. 
Rule.—If the numbers be greater than unity, add their 
logarithms together, and the fum is the logarithm of the 
produft. 
Ex. 1. What is the produEl of 2-71 X 38 X 49 ? 
0-4329693 log. of - - - 2 71 
1-5797836- 38 
1-6901961 - -40 
3-702949° 
5046-02 Produfh 
If any of the numbers be below unity, the rule is the 
fame, if you write their logarithms with the negative in¬ 
dex ; for then you have the true logarithm. 
Ex. 2. What is the produEl of 784 X 0-000079 X ° , ooooo36? 
2-8943161 log. of - - 784 
{8976271-.-• - - 0-000079 
~^' 5 S^ 3 02 5 ■ - - 0-0000036 
T‘3482457 
0-0000002229696 Produft. 
Here, we carry 2 from the decimal part of the logarithm 
(which is pofitive) to the index, which added to 2 (the 
firft index) make 4 for the pofitive part of the index, and 
the negative partis 11; therefore the fum is 4 — 11=_7- 
and there are fix ciphers before the firft fignificant figure! 
If the operation be performed by adding 10 to the in-! 
dex of the logarithm of the decimal factors, and there be 
r fuch faftors, you muft fubtraft ior from the index af¬ 
ter the addition. We will take the laft Example. 
2-8943161 log. of 784 
5-8976271 - - - 0-000079 
4-5563025 - - 0 0000036 
7-3482457 the fame as before. 
Here, the index of the fum is 13, and therefore 
10r = 20 ; hence, the index is 13 — 20 = —. 7. 
If the numbers be, fome or all of them, lefts than unity*., 
the product may be found" by the following Rule. Con-” 
fider the fignificant figures of each fa&or as whole num¬ 
bers 5 add together their logarithms; and fubtraft as many 
units from the index as there are decimal figures in all the 
numbers, and you have the true logarithm of the product 
For, let the given numbers be a, b, c, See. and fuppofe 
them to contain n decimals ; and let A, B^C, See-, be their 
refpedlive values, conceiving the fignificant figures to re¬ 
prefent whole numbers; then, by the nature of decimals, 
a X b X c x See. —-; hence, the loga¬ 
rithm of a X b x c X &c. — log. A +;Iog. B -f- log. C^- 
&c. — n. For the log. of io” is n. 
Ex. 1. What is the produEl of 2-4 X 0*007 X 0*54? Here, 
n = 6 ; hence. 
1-3802112 log. of 
24 
0-8450980 T 
- - - 7 
I- 7323938 . 
- 54 - 
• 3-9577030 --- 
- - 0-009072 Produdd. 
Here the index of the fum is 3, from which fubtrafl 6s-,, 
and the remainder is —3; therefore the. firft fignificant 
figure is 3 places below unity.. 
Ex* 
