LOGARITHMS.

IN order to abbreviate the tedious operations of .multiplication and division with large numbers, a series of numbers, called logarithms, were invented by Lord Napier, Baron of Marchinston in Scotland, and published in Edinburgh in 1614; by means of which the operation of multiplication may be performed by addition, and division by subtraction; numbers may be involved to any power by simple multiplication, and the root of any power extracted by simple division, _ .

In Table XXVI. are given the logarithms of all numbers from 1 to 9999; to each one ought to be prefixed an index, with a period or dot to separate it from the other part, as in decimal fractions; the numbers from 1 to 100 are published in that tableJwith their indices; but from 100 to 9999 the index is left out for the sake of brevity, but it may be supplied by this general rule, viz. the index of the logarithm of any integer, or mixed number, u ahcaus one less than the number of integer places in the natural number. Thus the index of the logarithm of any number (integer or mixed) between 10 and 100 is 1, from 100 to 1000 it is 2, from 1000 to 10000 is 3, &ic. the method of finding the logarithms from this table will be evident from the following examples. To find the logarithms of any number less than 100.

Rule. Enter the first page of the bible, and opposite the given number will be found the logarithm with its index prefixed.

Thus, opposite 71 is 1.85126, which is irs logarithm.

To find the logarithm of any number between 100 and 1000.

Ri'LE. Find the given number in file left hand column of the table of logarithms, and immediately under o in the next column, is a number, to which must be prefixed the number 2 as an index (because the number consists of three places of figures) and you will have'the sought logarithm.

Thus, if the logarithm of 649 was required ; this number being found in the left hand column, against it in the column marked 0 at the top (or bottom) is found 81224>'to w hich prefixing the index 2, we have the logarithm of 649=2.81224. •

To find the logarithm of any number between 1000 and 10000.

Rule. Kind the three left hand figures of the given number, in the left hand column of the table of logarithms, opposite to which, in the column that is marked at the top (or bottom) with the fourth figure, is to be found the sought logarithm; to which must be prefixed the index 3, because the number contains 4 places of figures. ,

Thus, if the logarithm of 649ft was required; opposite to 649, and in the column marked 5 at the top (or bottom) is 81258, to which prefix the index ff and we have the sought logarithm 3.81258.

To find the logarithm of any number above 10000.

Rule. Find the three first figures of the given number, in the left hand column of the table, and the fourth figure at the top or bottom, and take out the corresponding number as in the preceding rule; take also the difference between this logarithm and the next greater,' and multiply it by the given number exclusive of the four first figures, cross off at the right hand of the product as many figures as you had figures of the given number to multiply by; then add the remaining left hand figures of this product to the logarithm taken from the table, and to the sum prefix an index equal to one less than the number of integer figures in the given number, and you.will have the sought logarithm.

Thus, if the logarithm of 6 t9."»7 w as required: opposite to 6 '»3 and under 3 is 81258, the difference betw een this and the next greater number 81265 is 7, this multiplied by 7 (the last figure of the given number) gives 4!), crossing off the right hand figure leaves 4.9 or 5 to be added to 81208, which makes 81263, to this prefixing the index 4, we have the sought logarithm 4.81263.

Again, if the logarithm of 6495738 was required: the logarithm corresponding to 649 at the left, and 3 at the top. is as in the last example 81253. the difference between this and the next greater is 7, which multiplied by 738 (which is equal to the given number excluding the four first figures) gives 5166, Crossing off the three right hand figures of this product (because the number 788 consists of three figures) we have the correction 5 to be added to 81258; and the index to be prefixed is 6 because the given number consists of 7 places of figures, therefore the sought logarithm is 6.812G3. To find the logarithm of any mixed decimal number.

Rule. Find the logarithm of the number as if it was an integer by the last rule, to which prefix the index of the integer part of the given number.

Thus, if the logarithm of the mixed decimal 649.5738 was required;— find the logarithm of 6495738 without noticing the decimal point; this, in the last example, was found to be 81263, to this we must prefix the index 2, corresponding to the integer part 649; the logarithm sought will therefore be 2.81263.

To find the logarithm of any decimal fraction less than unity.

The index of-the logarithm of any number less than unity is negative, but to avoid the mixture of positive and negative quantities, it is common to borrow 10 or 100 in the index, which must afterwards be neglected in summing them with other indices; thus instead of writing the index — 1, it is generally written + 9 or + 99; but in general it is sufficient to borrow 10 in the index, and it is what we shall do in the rest of this work. In this way we may find the logarithm of any decimal fraction by the following rules.

Rule. Find the logarithm of a fraction as if it was a whole number;—see how many ciphers precede the first figure of the decimal fraction, subtract that number from 9 and the remainder will be the index of the given fraction.

Thus the log. of 0,0391 is 8.59218; the log. of 0,25 is 9.39794; die log. of 0,0000025 is 4.39794, &IC.

To find the logarithm of a vulgar fraction.

Rule. Subtract the logarithm of the denominator from the logarithm of the numerator (borrowing 10 in the index when the denominator is the greatest) the remainder will be the logarithm of the fraction sought.

EXAMPLE I. Required the log. of f?

From log. of 3 0.477131

Take log. of 8 . 0.903091

EXAMPLE H.
Required the log. of Si or't3
From log. of 13 1.11394
Take log. of 4 0.60306

Bern. log. | or, ,375 . 9.57403|Rem. log. of 31 or 3,25 0.51183

To find the number corresponding to any logarithm.

Rule. In the column marked 0 at the top (and bottom) of the table, seek for the next less logarithm, neglecting the index ; note the number against it, and carry your eye along that line until you find the nearest less logarithm to the given one, and you will have the fourth figure of the given number at the top, which is to be placed to the right of the three other figures; if you wish for greater accuracy, you must take the difference between this tabular logarithm and the next greater, also the difference between that least tabular logarithm and the given one; to the latter difference annex 2 or more ciphers, at the right hand, and divide it by the former difference, and place .the quotient* to the right hand of the four figures already found; and you will have the number sought expressed in a mixed decimal, the integer part of which will consist of a number of figures (at the left hand) equal to the index of the logarithm increased by unity, t

Thus, if the number corresponding to the logarithm 1.52634 was required; I look for 52634 in the column marked 0 at the top or bottom, and find it standing opposite to 336; now the index being 1, the sought number must consist of two integer places, therefore it is 33,6.

* This quotient must consist of as many places of figures as there were tinners annexed, conformable to the rules of the dirision of decimals. Thus, if the divisqr was 40, and the number to which two ciphers were annexed was S, making 2,00, the quotient must not be estimated as 5, but as M, and then two figures must be placed to the right of the four figures before found.

t If the index corresponds to o fraction less than unity, you must place as many ciphers to the left of that number as are equal to the index subtracted from S, the decimal point being placed lo the left of these ciphers', in this manner you will obtain the sought number.

If the given logarithm was 2.32838; I find that 32838 stands in the column marked 0 at the top or bottom, directly opposite to 213 which is the number sought, because the index being 2, the number must consist of S places of figures.

If the number corresponding to the logarithm 2.07345 was required; I look in the column 0, and find in it, against the number 374, the logarithm 57287, and guiding my eye along that line, I find the given logarithm 57345 in the column marked 5; therefore the mixed number sought is 3745, and since the index is 2, the integer part must consist of 3 places, therefore the number sought is 374,5. If the index had been 1, the number would have been 37,45; and if the index had been 0, the number would have been 3,745. If the index had been 8 corresponding to a number less than unity, the answer would have been 0,03745, Sic.

Again, if the number corresponding to the logarithm 5,57811 was required; I look in the column 0, and find in it against 378, and under 5, the logarithm 57807, the difference between this and the next greater logarithm 57818 being 11, and the difference between-57807 and the given number 57811 being 4, to this 4 I affix two ciphers, which make 400, and divide it by 11, the quotient is 36 nearly; this number connected with the former four figures make 378536, which is the number required, since the index being 5f the number must consist of six places of figures.

MULTIPLICATION BY LOGARITHMS. Rule. Add the logarithms of the two numbers to be multiplied, and the sum will be the logarithm of their product.

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Product 0,00815 log. 7.91116| Product 0,00075 log. 6.87506

In the last example the sum of the two indices is 16, but since 10 was borrowed in each number, I have neglected to in the sum, and the remainder 6 being less than the other 10, is evidently the index of the logarithm of a fraction less than unity

DIVISION BY LOGARITHMS. Rule. From the logarithm of the dividend subtract the logarithm of the divisor, the remainder will be the logarithm of the quotient.

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: 35 log.

EXAMPLE III. Divide 0,00315 by 0,0025. 0,00815 log. 0,0025 log.

Quotient 3,26 log.

1.54407

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EXAMPLE II.
Divide 40,32 by 22,4.
40,32 log.
22,4 log.

Quotient 1,8 log.

EXAMPLE IV. Divide 0,00075 by 0,025. 0,00075"log. •• 0,025 log.

1.60552 1.35025

0.25527

6.87506 8.39794

0,51322] Quotient 0,03 log. 8.47712 In Example III. both the divisor and dividend are fractions less than unity, and the divisor is the least, consequently the quotient is greater than unity. In Example IV. both fractions are less than unity, and since the divisor is the greatest, its logarithm is greater than that of the dividend; for that reason it was necessary to borrow 10 in the index previous to making the subtraction, hence the quotient is less than unity.

INVOLUTION BY LOGARITHMS.

Rule. Multiply the logarithm of the number given, by the index of the power to which the quantity is to be raised, the product will be the logarithm of the power sought. But in raising the powers of any decimal fraction it must be observed, that the first significant figure of the power must be put as many places below the place of units as the index of its logarithm wants of 10 multiplied by the index of the power.

EXAMPLE I. Required the square of IS? 18 log. .'

Answer 324 log.

EXAMPLE III. Required the square of 6,4? 6,4 log.

Answer 40,96 log.

2.51054

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EXAMPLE II.
Required the cube of 13?

1.255271 13 log. 1.11394.

"2| •' 3

Answer 2197 log. 3.34182

EXAMPLE IV. Required the cube of 0,25? 0,25 log.'

Answer 0,015625.

9.39794

3

1.61236] Answer 0,015625. 23.19392 In the last example the index 28 wants 2 of 30 (the product of 10 by the power 8) therefore the first significant figure of the answer, viz. 1, is placed two figures distant from the place of units.

EVOLUTION BY LOGARITHMS.

Rule. Divide the logarithm of the number by the index of the power, the quotient will be the logarithm of the root sought. But if the pow:er whose root is to be extracted is a decimal fraction less than unity, prefix to the index of its logarithm a figure less by one than the index of the power," and divide the whole by the index of the power, the quotient will be the logarithm of the root sought. ■ .

EXAMPLE I.

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What is the square root of 324?

324 log. 2)2.51055|

Answer 18 log.

EXAMPLE III. Required the square root of 40,96?

40,% log. 2)1.61236 Answer 6,4

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EXAMPLE II.
Required the cube root of 2197? .

2197 log. 3)3.34183 Answer 13 log. 1.11394

EXAMPLE IV.
Required the eube root of 0,015625?

0,015625 log. 8.19382 Prefix 2 to the index 3)28.19332 Answer 0,25 log. 9.39794

TO WORK THE RULE OF THREE BY LOGARITHMS.

When three numbers are given to find a fourth proportional in arithmetic, we make a statement and say, as the first number is to the second so is the third to the fourth; and by multiplying the second and third together, and dividing the product by the first, we obtain the fourth number sought. To obtain the same result by logarithms, we must add the logarith.-ts of the second and third numbers together, and from the sum subtract the logarithm of the first number, the remainder will be the logarithm of the sought fourth number.

EXAMPLE I.
If 6 yards of cloth cost 5 dollars, what
will 20 yards cost?
As 6

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log.

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log. 1.30103

2.00000
0.77815

log. I.221S5 The answer therefore is 16 dollars and 67-100ths or 16 dollars and 67 cents.'

EXAMPLE II. If a ship sails 20 miles in 7 hours, how much will she sail in 21 hours at the same rate? As 7

Is to 20

So is 21

Sum of M. and M.
Subtract the first
To 60

. The answer is

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* In this rule it is supposed that 10 was boa-owed in finding the index of the decintal'iicconUng'i'o llic rule page 30.

. To calculate COMPOUND INTEREST by Logarithm.

To' 100 dollars add its interest for one year; find the logarithm of this sum and reject 2 in the index, then multiply it by the number of years and parts of a year, for which the interest is to be calculated; to the product add the logarithm of the sum put at interest; the sum of these two logarithms will be the logarithm of the amount of the given sum for the given time.

. EXAMPLE.

Required the amount of the principal and interest of 305 dollars, let at ti per cent, compound interest for 7 years?

Adding 6 to 100 gives 106, whose logarithm, rejecting 2 in the index, is 0.0253]

Multiplied by 7

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Therefore the amount of principal and interest is 533 dollars and 83 cents. To find the Logarithm-sine, Tangent, or Secant, corresponding to any number of Degrees and Minutes, by Table XXVII.

The given number of degrees must be found at the bottom of the page when between 45° and 135°, otherwise at the top, the minutes being found in the column marked M, which stands on the side of the page on which the degrees are marked; thus, if the degrees are less than 45, the minutes are to be found in the left hand column, kc. and it must be noted that if the degrees are found at the top, the names of hour, sine, co-sine, .tangent, kc. must also be found at the top; and if the degrees are found at the bottom, the names sine, cosine, kc. must also be found at the bottom. Then opposite to the number of the minutes will be found the log-sine, log-secant, kc. in the'column marked sine, secant, kc. respectively.

EXAMPLE I.

Required the log. sine of 28° 37'?

Find 28° at the top of the page, directly below which, in the left hand column, find

EXAMPLE II.

Required the log. secant of 126° 20'.' •

Find" 126° at the bottom of the page, directly, above which, in the left hand column.

37'; against which in the column markedjfind 20'; against which, in the column mark sine is 9.6S029, the log. xine of the given'ed secant, is 10.22732 required, number of degrees; and in the same nian-j ner the tangents, kc. arc found.

To find the Logarithm-sine, Co-sine, Sfc. for Degrees, Minutes, and Seconds.

by Table XXVII.

Find the numbers corresponding to the even minutes next above and below the given degrees and minutes, and take their difference; then say, as (>Q" is to the number of seconds in the proposed number, so is that difference to a correction to be applied to the number corresponding to the least number of degrees and minutes; additive if it is the least the two numbers taken from the table, otherwise subtractiye.

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If the given seconds be }, J, \, or or any other even parts of a minute, the like parts may be taken of the difference of the logarithms, and added or subtracted as above, which may be frequently done by inspection.

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