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6. If in the equation ar=n, both members be raised to the mth power, amt=n". Here, ma is the logarithm of nm; from which it appears, that the logarithm of the power of any number, is equal to the logarithm of that number, multiplied by the index of that power.
7. If the mth root of both members of the equation at =n,
be taken, then, am=nm; but is the logarithm of nm; from which it appears, that the logarithm of the root of any number, is equal to the logarithm of that number divided by the index of the root.
8. It is evident, that the results obtained in the last four artieles are equally true, whether the logarithms be positive or negative. These results show, that the addition of logarithms corresponds to the multiplication of their numbers; the subtraction of logarithms, to the division of numbers; their multiplication, to the raising of powers; and their division, to the extraction of roots.
9. Returning to the equation a+=n, in which Ex=log. 1, and applying it to the common system, in which the base is 10, we have, (10)4 : (10)3: (10)2 : (10)':(10)6 : (10)–1:(10)-2: (10)-3: (10) 10000 : 1000 : 100 : 10
: 0.01 :0.001 -: 0.0001 num 0:-1
log. Unity being the number whích divides the whole numbers from the decimal fractions, we shall begin with it, and explain some properties of the logarithms of whole numbers. The logarithm of 1 is 0; and this is the case in all systems, for whatever be the base, its O power is 1: but the index of the base is the logarithm of the power; therefore, 0 is the logarithm of 1. As the logarithms increase with the numbers from unity upwards, the logarithms of all numbers, which are greater than 1, and less than 10, are greater than 0, and less than 1: their values are generally expressed by decimal fractions ; thus, the log. 2=0.301030. The logarithms of numbers greater than 10, and less than 100, lie between 1 and 2, and are generally expressed by unity and a decimal fraction: thus, the log. 50=1.698970.
The logarithms of numbers greater than 100, but less than
by uniting 2 with a decimal fraction: thus, the log. 126= 2.100371. The whole number on the left of the decimal point is called the characteristic, or index of the logarithm. The number of units which it contains, is always one less than the number of places of figures in the number whose logarithm is taken. Thus, in the first case, for numbers between 1 and 10, there is but one place of figures, and the characteristic is 0. In the second case, for numbers between 10 and 100, there are two places, and the characteristic is 1. In the third case, for numbers between 100 and 1000, there are three places, and the characteristic is 2; and in like manner for any number of places whatsoever.
TABLE OF LOGARITHMS.
10. If the logarithms of all the numbers between 1 and any given number, be calculated and arranged in a tabular form, such table is called a table of logarithms. The table annexed shows the logarithms of all numbers between 1 and 10,000.
11. The first column, on the left of each page of the table of logarithms, is the column of numbers, and is designated by the letter N; the logarithms of these numbers are placed directly opposite them, and on the same horizontal line.
12. To find, from the table, the logarithm of any whole number.
If the number be less than 100, look on the first page of the table of logarithms, along the columns of numbers under N, until the number is found; the number directly opposite it, in the column designated Log., is the logarithm sought.
13. When the number is greater than 100, and less than 10,000.
Find, in the column of numbers, the first three figures of the given number. Then, pass across the page, in a horizontal line, into the columns marked 0, 1, 2, 3, 4, &c., until you come to the column which is designated by the fourth figure of the given number: to the four figures so found, two figures taken from the column marked 0, are to be prefixed. If the first four figures found stand opposite to a row of six figures in the column marked 0, the two figures from this column, which are left hand; but, if the first four figures are opposite a line of only four figures, you are then to ascend the column, till you come to the line of six figures : the two figures at the left hand are to be prefixed, and then the decimal part of the logarithm is obtained; to which prefix the characteristic (9), and you
have the logarithm sought. In several of the columns designated 0, 1, 2, 3, 4, 5, &c., small dots are found.
In such cases, a cipher must be written for each of those dots; and the two figures, from the first column, which are to be prefixed, are found in the horizontal line directly below. Thus, the log. 2188 is 3.340047, the two dots being changed intố two ciphers, and the 34 from the column 0, prefixed.' The two figures from the column 0, must also be taken from the line below, if any dots” shall have been passed over, in passing along the horizontal line: thus, the logarithm of 3098 is 3.491081, the 49 from the coluinn 0 being taken from the line 310.
14. If the number exceeds 10,000, or consists of five or more places of figures, consider all the figures after the fourth from the left hand, as ciphers. Find, from the table, the logarithm of this number, which will be the same as the logarithm of the first four places, excepting the characteristic. Take from the last column on the right of the page, marked D, the number on the same horizontal line with the logarithm, and multiply this number by the numbers that have been considered as ciphers: then, cut off from the right-hand as many places for decimals as there are figures in the multiplier, and add the product, som obtained, to the first logarithnı, for the logarithm sought.
Let it be required to find the logarithm of 672887. The log. of 672800 is found; on the 11th page of the table, to be 5.827886, by prefixing the characteristic 5. The number corresponding in the column D is 65, which being multiplied by 87, the figures regarded as ciphers, gives 5655; then, pointing off two places for decimals, the number to be added is 56.55. This number being added to 5.827886, gives 5.827942 for the logarithm of 672887; the decimal part, .55, being omitted.
This method of finding the logarithms of numbers from the table, supposes that the logarithms are proportional to their respective numbers, which is not rigorously true. In the example, the logarithm of 672800 is 5.827886; of 672900,
logarithms is 65. Now, as 100, the difference of the numbers, is to 65, the difference of their logarithms, so is 87, the difference between the given number and the least of the numbers used, to the difference of their logarithms, which is 56.55: this difference being added to 5.827886, the logarithm of the less number, gives 5.827942 for the logarithm of 672887. The use of the column of differences is therefore manifest.
15. The logarithm of a fractional number is casily found, from what has already been said. "If the fractional number exceeds unitý, as , its logarithm is equal to the log. 136-log. 25 (5). If it be less than unity, as , its logarithm may be written under two different forms. First, the log in log. 15 – log. 125-=-(log. 125_log. 15)=-(2.096910 1.176091)=-0.920819; the number 0.920819 being entirely negative. In the equation log. 15-log. 125= - 0.920819, if the log. 125 be transposed to the second member, the log. 15 =log. 125-0.920819. Let N' be the number whose loga: rithm is ---0.920819, and N the number whose logarithm is +0.920819; then, the log. 15-log. 125=log. N. Since the difference of logarithms of the two numbers is equal to the
125 logarithm of their quotient (5), the log. 15=log. But if the logarithms are equal, the numbers themselves are equal; 125 15 1
15 therefore, 15=
since is the number 125 N
125 whose logarithm is -0.920819. As the same reasoning holds true for any numbers whatever, we conclude, that the number answering to a negative logarithm, is the reciprocal of the number answering to this same logarithm regarded as positive.
16. To find the logarithm of a proper fraction under another form. Let the fraction be N=zki.
Let this fraction be multiplied by 10, 100, 1000, 10,000, or such higher power of 10, as to make it greater than unity. If it be multiplied by
10,000 X 125 10,000, we shall have, 10,000N=
and taking the
5627 * logarithms, 4+log. N=4+log. 125 – log.5627=4+2.096910
-3.750277 = 6.096910 -3.750277=2.346633: hence the
the characteristic only, and not to the decimal part of the logarithm. In such case, the minus sign is written above the number; thus, 2. If, then, it be required to express the logarithm of a fractional number, under such a form that the characteristic only shall be negative, add such a whole number to the logarithm of the numerator, as will make it greater than the logarithm of the denominator; from this sum, subtract the logarithm of the denominator, and from the remainder, the whole number which was added to the logarithm of the numerator : the remainder is the logarithm sought.
17. To find the logarithm of a decimal number. If the number be composed of a whole number and a decimal, such as, 36.78, it may be put under the form 479: the log. 17= log. 3678—2=3.565612—2=1.565612; from which we see, that the mixed number may be treated as a whole number, except in fixing the value of the characteristic, which is one less than the number of places on the left of the decimal point.
18. The logarithm of a decimal fraction is also readily found. The log. 0.8=log. 1=log. 8—1=-1+log. 8. . Now the log. 8 is 0.903090, which is positive, and less than 1; hence log. 0.8=1:903090, where the minus sign belongs to the characteristic only (16); hence, it appears, that the logarithms of tenths, are the same as the logarithms of the corresponding whole 'numbers, excepting, that the characteristic instead of being 0, is
-1. If the fraction were of the form .06, it might be written 14o; taking the logarithms, log. =log. 06-2=-2+ log. 06. Now, the log. 06 is but the log. 6; therefore, the log. .06=2.778151, the minus sign belonging only to the characteristic, the decimal part being positive (16). If the decimal were .006, its logarithm would be the same, excepting the characteristic, which would be — 3. It is, indeed, evident, that the negative characteristic will always be one greater than the number of ciphers between the decimal point and the first significant place of figures ; therefore, the logarithm of a decimal fraction is found, by considering it as a whole number, and then prefixing to its logarithm a negative characteristic, greater by unity than the number of ciphers between the decimal point and the first significant place of figures.