denotes, that that is negative, while the decimal part of the logarithm is positive. 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 index 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 error 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. 2.) In Briggs' system, the logarithm of 10 is 1. (Art. 5.) 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 index 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 loga rithms 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, considered as impossible. But as there is no difference in the multiplication, division, involution, &c. of positive and negative quantities, except in applying the signs; they may be considered as all positive, while these operations are performing by means of logarithms; and the proper signs may be afterwards affixed. 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, a=the least term, L-its logarithm, r=the ratio, 1 its logarithm ; Then the logarithm of ar is L+1, (Art. 1.) of ar 2 is L+21, of ar3 is L+31, &c. Here, the quantities a, ar, ar3, ar, &c. are in geometrical progression. (Alg. 436.) And their logarithms L, L+1, L+21, L+31, &c. are in arithmetical progression. (Alg. 423.) * See note C. Omit 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 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, &c. 21. The logarithms of numbers less than a unit are negative. (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. (Älg. 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, y=the corresponding ordinate. 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. 2 |