denotes, that that is negative, while the decimal part of the logarithm is positive. of 0.3, is 1.47712, The logarithm of 0.06, is 2.77815, (of 0.009, is 3.95424, And universally, 11. The negative index of a logarithm shows how far the first significant figure of the natural number, is removed from the place of units, on the right; in the same manner as a positive index shows how far the first figure of the natural number is removed from the place of units, on the left. (Art. 8.) Thus in the examples in the last article, The decimal 3 is in the first place from that of units, 6 is in the second place, 9 is in the third place; And the indices of the logarithms are 1, 2, and 3. 12. It is often more convenient, however, to make the inder of the logarithm positive, as well as the decimal part. This is done by adding 10 to the index. This is making the index of the logarithm 10 too great. But with proper caution, it will lead to no errour in practice. 13. The sum of the logarithms of two numbers, is the logarithm of the product of those numbers; and the difference of the logarithms of two numbers, is the logarithm of the quotient of one of the numbers divided by the other. (Art. 1.) In Briggs' system, the logarithm of 10 is 1. (Art. 3.) If therefore any number be multiplied or divided by 10, its logarithm will be increased or diminished by 1: and as this is an integer, it will only change the inder of the logarithm, without affecting the decimal part. Thus the logarithm of 4730 is 3.67486 The logarithm of the product 10 is 1. 47300 is 4.67486 And the logarithm of the quotient 473 is 2.67486 Here the index only is altered, while the decimal part remains the same. We have then this important property, 14. The DECIMAL PART of the logarithm of any number is the same, as that of the number multiplied or divided by 10, 100, 1000, &c. This property, which is peculiar to Briggs' system, is of great use in abridging the logarithmic tables. For when we have the logarithm of any number, we have only to change the index, to obtain the logarithm of every other number, whether integral, fractional, or mixed, consisting of the same significant figures. The decimal part of the logarithm of a fraction found in this way, is always positive. For it is the same as the decimal part of the logarithm of a whole number. 15. In a series of fractions continually decreasing, the negative indices of the logarithms continually increase. Thus In the series 1, .1, .01, .001, .0001, .00001, &c. The logarithms are 0, -1, -2, -3, -4, -5, &c. If the progression be continued, till the fraction is reduced to 0, the negative logarithm will become greater than any assignable quantity. The logarithm of 0, therefore, is infinite and negative. (Alg. 447.) 16. It is evident also, that all negative logarithms belong to fractions which are between 1 and 0; while positive logarithms belong to natural numbers which are greater than 1, As the whole range of numbers, both positive and negative, is thus exhausted in supplying the logarithms of integral and fractional positive quantities; there can be no other numbers to furnish logarithms for negative quantities. On this account, the logarithm of a negative quantity is, by some writers, said to be impossible. It appears to be more proper, however, to consider the logarithms of negative quantities, as being the same with the logarithms of positive quantities. Logarithms are the exponents of powers and roots. (Art. 2.) But an exponent may be applied to a negative power or root, as well as to a positive one. arithm of Thus the cube of a is -a3 And the cube of +a is +a3) Alg. 218. Though these two powers are one positive, and the other negative, yet their exponents are the same. So also, the loga, or of any power of a, is the same as the logarithm of the same power of -a. Positive and negative quantities, therefore, can not be distinguished from each other by their logarithms. 17. If a series of numbers be in GEOMETRICAL progression, their logarithms will be in ARITHMETICAL progression. For, in a geometrical series ascending, the quantities increase by a common multiplier; (Alg. 436.) that is, each, succeeding term is the product of the preceding term into the ratio. But the logarithm of this product is the sum of the logarithms of the preceding term and the ratio; that is, the logarithms increase by a common addition, and are, therefore, in arithmetical progression. (Alg. 422.) In a geometrical progression descending, the terms decrease by a common divisor, and their logarithms, by a common difference. Thus the numbers 1, 10, 100, 1000, 10000, &c. are in geometrical progression. And their logarithms 0, 1, 2, 3, 4, &c. are in arithmetical progression. Universally, if in any geometrical series, ar, Here, the quantities a, ar2, ar 3, in geometrical progression. (Alg. 436.) ar1, &c. are And their logarithms L, L+1, L+21, L+31, &c. are în arithmetical progression. (Alg. 423.)* 18. Hyperbolic logarithms. Although the system of logarithms which has now been explained, is far more convenient than any other, for the common purposes of calculation, and is the only one in general use, at the present time; yet it is not that which was first proposed, by the celebrated inventor of logarithms Lord Napier. For particular reasons, he made the radix of his system 2.718281828459, instead of 10. This produces a change throughout the whole series, except the logarithm of 1, which, in every system, is 0. Napier's logarithms are also called hyperbolic logarithms, from certain relations which they have to the spaces between the asymptotes and the curve of an hyperbola; although these relations are not, in fact, peculiar to Napier's system. These logarithms have some particular uses, to which the common tables are not adapted. THE LOGARITHMIC CURVE. 19. The relations of logarithms, and their corresponding numbers, may be represented by the abscissas and ordinates of a curve. Let the line AC (Fig. 1.) be taken for unity. Let AF be divided into portions, each equal to AC, by the points 1, 2, 3, &c. Let the line a represent the radix of a given system of logarithms, suppose it to be 1.3; and let a2, a3, &c. correspond, in length, with the different powers of a. Then the distances from A to 1, 2, 3, &c. will represent the logarithms of a, a2, a3, &c. (Art. 2.) The line CH is called the logarithmic curve, because its abscissas are proportioned to the logarithms of numbers represented by its ordinates. (Alg. 527.) 20. As the abscissas are the distances from AC, on the line * See note C. It may not be correct to ascribe to Napier the original discovery of the principle of logarithms. But he was the first who constructed logarithmic tables, and adapted them to the methods of computation to which they are now so extensively applied. These tables were published in 1614. For a particular History of logarithms, see the Introduction to Hutton's Mathematical Tables. AF, it is evident, that the abscissa of the point C is 0, which is the logarithm of 1=AC. (Art. 2.) The distance from A to 1 is the logarithm of the ordinate a, which is the radix of the system. For Briggs' logarithms, this ought to be ten times AC. The distance from A to 2 is the logarithm of the ordinate a2; from A to 3 is the logarithm of a3, 21. The logarithms of numbers less than a unit are nega tive. (Art. 9.) These may be represented by portions of the line AN, on the opposite side of AC. (Alg. 507.) The ordinates a-1, a-2, a-3, &c. are less than AC, which is taken for unity; and the abscissas, which are the distances from A to -1, -2, -3, &c. are negative. 22. If the curve be continued ever so far, it will never meet the axis AN. For, as the ordinates are in geometrical progression decreasing, each is a certain portion of the preceding one. They will be diminished more and more, the farther they are carried, but can never be reduced absolutely to nothing. The axis AN is, therefore, an asymptote of the curve. (Alg. 545.) As the ordinate decreases, the abscissa increases; so that, when one becomes infinitely small, the other becomes infinitely great. This corresponds with what has been stated, (Art. 15.) that the logarithm of 0 is infinite and negative. 23. To find the equation of this curve, Let a=the radix of the system, x=any one of the abscissas, Then, by the nature of the curve, (Art. 19.) the ordinate to any point, is that power of a whose exponent is equal to the abscissa of the same point; that is (Alg. 528.) y=a*.* *For other properties of the logarithmic curve, see Fluxions. |
