LOGARITHMS. 
Ex. What is the i6tk power of 1-05 ? 
0 0211893 log. of X - 05. 
16 
0-3390288 -- 2-182875 the power required. 
If the number to be railed to a power be lefs than unity, 
and you exprefs its logarithm b}' a negative index, the ope¬ 
ration is performed by the following Rule : Multiply the 
index of the logarithm, and the decimal part, feparately by 
the index of the power, and fubtraft the former product 
(it being negative) from the latter, and the remainder is 
the logarithm of the power. For, the index being nega¬ 
tive, and the decimal part pofitive, the logarithm, as it 
fo (lands, cannot generally be multiplied as one quantity; 
the index and logarithm ape therefore multiplied feparately, 
and the former produft is taken from .the latter. 
Ex. 1. What is the 8-3 power of 0-04.2 ? 
The log. of 0-042 is “ 6232493 ; hence, 
’6232493 
8 3 
18697479 
49859944 
From 5’i72969i9 
Sub. i6 - 6 =8-3 X 2 
Rem. 1 2 ''572969i.9 log. of 0-0000000000037408 the power 
required. 
Ex. 2. What is the 0-07 power of 0-00563 ? 
The log. of 0-00563 is T'75°5°84 ; hence, 
•7505084 
o-0 7 
From 0-052535588 
Sub. 0.21 _=20-07 X 3 
Rem. T 842535588 log. of 0-695882 the power required. 
If you reduce tire logarithm of the root to a negative lo¬ 
garithm, the operation may be performed by one multi¬ 
plication. Let us take the cafe of the laft example. 
The logarithm 3"7505084 = —2-2494916 ; hence, 
—• 2-2494916 
0-07 
— 0-157464412 = -5-842535588 the fame as before. 
The other method may generally be found mod ready in 
praftice, as you ha've here to reduce the given logarithm 
to a negative logarithm, and then at laft to reduce it back 
again. This method, however, will be found very ufe- 
ful, when you have to find the value of a decimal num¬ 
ber having an index expreffed by a vulgar f raft ion. 
If the number to be raifed to a power be lefs than unity, 
and you exprefs its logarithm by adding 10 to the index, 
the operation may be performed by the following Rule : 
Let n be the index of the power; multiply the logarithm 
of the root by n, and from the produft fubtraft ion, and 
the remainder is the true logarithm of the power. For, 
the logarithm of the root being 10 greater than the true 
logarithm, when you multiply it by n, the logarithm mull 
be ion greater than the true logarithm, and therefore you 
inuft fubtraft ion from the produft, in order to get the 
true logarithm of the power. 
Ex. 1. What is the 8-3 power of 0-042 ? 
8 6232493 log. of 0-042 
_ 8-3 
258697479 
689859944 
From 71-57296919 
Sub. 83- = 10 x 8-3 
Rem. iT'57296919 log. of 0-0000000000037408 the power 
required. 
Vol. XII. No. 881. 
$01 
Ex. 2. What is the o'oj power rf 0-00563 ? 
7-7505084 log. of 0-00563 
0-07 
From o'542535588 
Sub. 0-7 =10x0-07 
Rem. T'842535588 log. of 0-695882 the power required. 
It will l'ometimes happen, that the logarithm produced 
will not he the exact logarithm of the number required. 
Forinftance; the log. of 7 s is 5 X log. 7 = 5 ;< 0-8450980 
= 4-2254900. Now 7 s — 16807, the log. of which is 
4-2254902, differing from the above log. by 2 in the lad; 
figure; and this ariles from the log. of every number being 
only an approximation ; on which account, the error, 
w-hen multiplied, becomes manifeft. Had the firft loga¬ 
rithm been continued to one or two more places, then, 
after the multiplication, 2 would have been carried to the. 
feventh place of decimals, and the produft would have 
agreed with the logarithm found in the Table. 
Evolution by Logarithms. 
Rule.—Divide the logarithm of the given number by 
the number denoting the root to be extracted, and the 
quotient is the logarithm of the root. 
Ex. What is the 5 tk root of 784? 
5)2-8943161 log. of - - 784 
0-5788632 ——— - - 3-792044 the root. 
If the given number be lefs than unity, and you exprefs 
its logarithm by a negative index, the operation is per¬ 
formed by the following Rule: Reduce the logarithm of 
the number to a negative logarithm, and then divide it by 
the number denoting the root to be extrafted, and you 
have the logarithm of the root expreffed by a negative lo¬ 
garithm ; reduce this back again to a logarithm with a ne¬ 
gative index and pofitive decimal part, and you have the 
true logarithm of the root. 
Ex. 1. What is the 0-72 root of 0-096 ? The logarithm of 
• J C '72 
o - oq6 is -r-9822712 = — 1-0177288; and-— 
y 2 v / _ ,. OI772 8g — 
— 1-4135122 = "2-5864878 the log. of 0-03859116 the root 
required. 
Ex. 2. What is the 504 root of 0-2? The log. of 
— 0-6989700 
0*2 is 1 -3010300 = — 0-6989700, and --- = 
5°4 
— 0-0013868 = T’9986i32 the logarithm of 0-9968119 the 
root required. 
Ex. 3. What is the o-i root of o-i ? The log. of o-i is 
T'ooooooo; and, as here is no decimal part, we have 
— 1 _ 
-^7^2=—10= 1 o "ooooooo, the logarithm of o-oooooooooi 
the root required. 
This Rule, for the extraction of roots of numbers lefs 
than unity, admits of no variety of cafes in praftice. 
To find the value of a quantity expreffed by a number whofe 
index is a vulgar fraction. —Rule. Multiply the logarithm 
of the number by the numerator of the fraftion denoting 
the index, and divide the produft by the denominator, 
and the quotient is the logarithm of the quantity required. 
But, it the given number be a decimal, reduce its loga¬ 
rithm to a negative logarithm ; multiply it by the numera¬ 
tor of the fraftion expretling the index, and divide the 
produft by the denominator; and reduce the negative lo¬ 
garithm back to the logarithm with a negative index 
only ; and you get the logarithm of the quantity required. 
Ex. What is the value 0-096|rj- ■ The log. of 0-096 is 
“-9822712= — 1-0177288; hence, 
— 1-01772885 
_ 5 
9) — 5-0886440 
— 0-5 654049 = T'434595D the logarithm of 0-2720164 
the quantity lought. 
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