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This number e is very important, and is called the base of the Naperian Logarithms, for reasons to be explained hereafter. We can casily ascertain its value in the following manner:-In the above series let 1. Then
The series, therefore, gives the numerical value of e. This value may be calculated as follows:
•718281827; and hence e27182818, which is quite accurate, so far as it goes. The student will observe, that the reasoning in the above article is founded upon the assumptions (1), that aa can be expanded in a series of ascending powers of 2; (2), that it can be expanded in only one series of that kind. These assumptions, in the present case, may be considered as resting on the fact that the expansion of a is simply a transformation of the binomial theorem. The same remark applies to the following article-If we make the assumption general, viz., that every function of a can be expanded in a single series of ascending powers of a, we enter upon a question which has given rise to many discussions, which cannot be further noticed here.
Definition. If ax, then y is called the logarithm of x to the base a, and is generally written y = loga x where loga means "logarithm to base a."
Now, since (1+x)" is identically the same as a", these two series must be identically
13. On the Calculation of the Arithmetical Values of Quantities expressed by Infinite Scries. In the Treatise on Elementary Algebra the method has been explained of obtaining in numbers the value of an algebraical expression, when definite values are assigned to the letters composing the expression. For instance, if a=2 and 6 = 5, then (2a +b) (b-a)=27. The student may ask, How can an infinite series be reduced? Although we have already given three instances of the manner of doing this, the question is well worth a distinct consideration. We have already seen that
Now, if x = 1, the fraction is equal to 2, and we know that if we took the whole number of terms of the series we should get exactly 2. The first two terms are 1.5 the first three 175; the first four 1875; the first five terms 1.9375; the first six 1.96875; each result being nearer to the truth than the one before. Thus, by taking a sufficiently large number of terms, we can get as near to the exact value as we like. The series, in fact, affords the means of approximating to the true value. Of course, in such a case as the above, we should not care for the approximation, since we can so readily get the real value. But in the large majority of cases we cannot get at the real value, or even the real value cannot be expressed by digits at all, i. e., is not commensurable with unity; in such cases, the approximate value is the only one we can get,
and the series is the means by which we get it. the Naperian logarithm e. But what is e?
For instance, we have called the base of It is such a number that
1 = (e — 1) — 1 (e − 1 )2 + } (e — 1)3 — &c.
This is an equation we have no method-no direct method-of solving. We have seen, however, that e is expressed by the series
From this, as we have already seen, we can find that e=2·7182818, &c. The student may think an approximation a very unsatisfactory result; but he must remember two things:-(1). That greater part of the quantities we have to deal with cannot be expressed in whole numbers, or in vulgar fractions, e. g., so common and elementary an expression as 2 cannot be expressed as a vulgar fraction; and (2), that in practice no measurement is accurate, but is known to lie within certain limits. For instance, if a tailor measures a piece of cloth, he calls it a yard, though it may happen to be a quarter of an inch more or less. In like manner the most refined scientific measurements (the length of the second's pendulum, of an arc of the meridian, &c.,) are generally the means of several results, and are accurate to within certain very small limits. Now, in approximating to a result by means of a series, we can always get to within any given limits that may be assigned. And thus approximations by means of series are as accurate as any, the most refined, measurements can be.
In practice, if we know a number to be true for the first six or seven places of decimals, it is generally known with sufficient or even more than sufficient accuracy. Thus, if we are certain that x. lies between 3.16754 and 3·16755, we may call it 3.16754; although if we calculated to a greater nicety we might obtain x = 3.1675438295, for the error we commit is 100000, e. g., an error less than of an inch in one mile.
It is to be observed that in calculating the value of a series we must calculate each term to one or two more places of decimals than the result we wish to obtain, so as to be quite sure that we carry the right number to the seventh place. Thus, in finding the value of e (Art. 11), we calculated each term to nine places of decimals, to ensure that our result should be true to seven places.
It is also to be observed that in cutting off the eight and subsequent decimal places, if the eighth place is 5, 6, 7, 8, or 9, we add 1 to the seventh place; but if 4, 3, 2, 1, or 0, we simply omit it. Thus we reckon 2.597163458272.5971635
For it is plain that 2:59716345827 is nearer to 2.5971635 than to 2.5971634: whereas, as in the second instance, the contrary is the case.
We now proceed to consider the subject of logarithms in detail.
14. To explain the principle on which Logarithms may be used to facilitate calculations. From the definition of a logarithm already given, it follows that if M= a* then ≈ is the logarithm of M to the base a; and if Nay then y is the logarithm of N to the base a. Now observe M X N = a*+, whence it is plain that the multiplication of one number by another corresponds to the addition of their logarithms. In like manner, M÷N=ay or the division of one number by another corresponds to the subtraction of the logarithm of the dividend from that of the divisor. Again, Mmamx, or the raising of a number to a given power corresponds to the multiplication of the logarithm by that power. In like manner Mm am or the extraction of the root of a
given number corresponds to the division of the logarithm by that root. So that if we knew the logarithm which corresponds to any number whatever, and wished to find the product of two numbers, we should merely have to write down the logarithms of the numbers, add them, and then the number whose logarithm is that sum will be the product of the two given numbers; and similarly for the other rules.
No, tables have been calculated which give us the logarithm corresponding to any numbe between 1 and 10,000,000. Hence, by using these tables properly, multiplication is performed by means of addition; and in like manner division by means of subtraction, involution by multiplication, and evolution by division.
In the following pages we shall first explain the method by which these tables are calculated, and then proceed to show how they are practically employed,
15. The following results follow manifestly from what has been said.
(1.) That if P = Q, then log.P= logaQ.
From this it follows that if we know the logarithm of a given number to a given base, we can find its logarithm to another base, by dividing the first logarithm by the logarithm of the new base; for instance, suppose our tables give the logarithms of num
bers to the base 10, and suppose we wished to find the logarithm of a given number (N) to the base 19, we have
Then log 19 and log1N are given by the tables, and therefore we know log19N by division.
17. The practical advantage of Calculating Logarithms to the Base 10..
The Tables of Logarithms commonly printed, are logarithms to the base of 10. In all future articles, whenever we write log x, we mean logarithm to the base 10.
We might calculate tables to the base e; and the calculation is obviously rendered much easier when this base is employed by the circumstance that
And, in point of fact, the inventor of logarithms, Napier, actually, calculated logarithms to this base e, which is hence called the base of the Napierian logarithms.
For the purposes of numerical calculation; however, the base 10 possesses the fol lowing decisive advantage over any other.
Then 10x+n NX 10"
Now, supposen to be a whole number, then N. X 10" has the same digits as N in the same order, and only differs from it in having its decimal point shifted n places to the right; and again—
N = 10n
And when n is a whole number,
only differs from N in having its decimal point
shifted n places to the left. It follows, therefore, that the decimal part of the logarithm of a number is the same wherever the decimal point may be in the number, and that for every place that the decimal point in the number is shifted to the right, 1. is added to the logarithm; and for every place, it is shifted to the left, 1. is subtracted from the logarithm. Thus the table gives us
It is plain, then, that one calculation gives us the logarithm of the above five numbers, and in fact of as many numbers as can be made by shifting the decimal point to different positions in the combination 75684; but if we adopted any other base, we should require a separate calculation for each of them. This advantage; which the base