this. If the radix is 10, as in the common system, every other number is to be considered as some power of 10. If the exponent is a fraction, and the numerator be increased, the power will be increased; but if the denominator be increased, the power will be diminished. 6. To obtain then the logarithm of any number, according to Briggs's system, we have to find a power or root of 10 which shall be equal to the proposed number. The exponent of that power or root is the logarithm required. Thus 7. A logarithm generally consists of two parts, an integer and a decimal. Thus the logarithm 2.60206, or, as it is sometimes written, 2+.60206, consists of the integer 2, and the decimal .60206. The integral part is called the characteristic or index* of the logarithm; and is frequently omitted, in the common tables, because it can be easily supplied, whenever the logarithm is to be used in calculation. By art. 3d, the logarithms of 10000, 1000, 100, 10, 1, .1, .01, .001, &c. are 4, 3, 2, 1, 0, -1, -2, -3, &c. As the logarithms of 1 and of 10 are 0 and 1, it is evident, that, if any given number be between 1 and 10, its logarithm will be between 0 and 1, that is, it will be greater than 0, but less than 1. It will therefore have 0 for its index, with a decimal annexed. Thus, the logarithm of 5 is 0.69897. * The term index, as it is used here, may possibly lead to some confusion in the mind of the learner. For the logarithm itself is the index or exponent of a power. The characteristic, therefore, is the index of an index. For the same reason, if the given number be between We have, therefore, when the logarithm of an integer or mixed number is to be found, this general rule: 8. The index of the logarithm is always one less, than the number of integral figures, in the natural number whose logarithm is sought: or, the index shows how far the first figure of the natural number is removed from the place of units. Thus, the logarithm of 37 is 1.56820. Here, the number of figures being two, the index of the logarithm is 1. The logarithm of 253 is 2.40312. Here the proposed number 253 consists of three figures, the first of which is in the second place from the unit figure. The index of the logarithm is therefore 2. The logarithm of 62.8 is 1.79796. Here it is evident that the mixed number 62.8 is between 10 and 100. The index of its logarithm must, therefore, be 1. 9. As the logarithm of 1 is 0, the logarithm of a number less than 1, that is, of any proper fraction, must be negative. Thus, by art. 3d, The logarithm of or .1 is -1, or .01 is -2, or .001 is -3, &c. 10. If the proposed number is between 10 and 100, its logarithm must be between -2 and -3. To obtain the logarithm, therefore, we must either subtract a certain frac tional part from -2, or add a fractional part to -3; that is, we must either annex a negative decimal to -2, or a pos of .008 is either -2.09691, or -3+90309.* The latter is generally most convenient in practice, and is more commonly written 3.90309. The line over the index 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. Thus, for 1, 9 is written, Because--1+10=9, for -2, 8, &c. 2+10=8, &c. * That these two expressions are of the same value will be evident, if we subtract the same quantity, +.90309 from each. The remainders will be equal, and therefore the quantities from which the subtraction is made must be equal. 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's 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 index of the logarithm, without affecting the decimal part. Thus, the logarithm of 4730 is 3.67486 And the logarithm of 10 is 1. The logarithm of the product And the logarithm of the quotient 47300 is 4.67486 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's 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. 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. 359.) 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. 326.) 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. See Note A. |