TRIGONOMETRY. BOOK I. LOGARITHMS. 1. THE LOGARITHM of a number is the exponent of the power to which a given fixed number must be raised in order to produce the first number. 2. The BASE of the system is the fixed number. 3. The base, in the common system of logarithms, is 10. Hence, since It thus appears that, in the common system, the logarithm of every number between 1 and 10 is some number between 0 and 1; that is, a proper fraction. The logarithm of every number between 10 and 100 is some number between 1 and 2; that is, 1 plus a fraction. The logarithm of every number between 100 and 1,000 is some number between 2 and 3; that is, 2 plus a fraction; and so on. 4. By means of negative exponents the application of logarithms may be extended, in the common system, to numbers less than 1. Thus, since From this it appears that the logarithm of every number between 1 and 0.1 is some number between 0 and -1; that is, -1 plus a fraction. The logarithm of every number between 0.1 and 0.01 is some number between 1 and -2; that is, -2 plus a fraction. The logarithm of every number between 0.01 and 0.001 is some number between -2 and -3; that is, -3 plus a fraction; and so on. 5. In the common system, as the logarithms of all numbers which are not exact powers of 10 are incommensurable with those numbers, their values can only be obtained approximately, and are expressed by decimals. 6. The integral part of any logarithm is called the CHARACTERISTIC, and the decimal part is sometimes called the MAN TISSA. 7. The characteristic of the logarithm of ANY NUMBER GREATER THAN UNITY, is one less than the number of integral figures in the given number. For it has been shown (Art. 3) that the logarithm of 1 is 0, of 10 is 1, of 100 is 2, of 1000 is 3, and so on. 8. The characteristic of the logarithm of ANY DECIMAL FRACTION is a negative number, and is one more than the number of ciphers between the decimal point and the first significant figure. For it has been shown (Art. 4) that the logarithm of 0.1 is -1, of 0.01 is -2, of 0.001 is -3, and so on. NOTE. In general, whether the given number be integral, fractional, or mixed, the characteristic of the logarithm of any number expressed decimally is the distance of the first, or left-hand, significant figure from the units' place, being positive when that figure is on the left of the units' place, and negative when on the right. GENERAL PROPERTIES OF LOGARITHMS. 9. The logarithm of a PRODUCT is equal to the sum of the logarithms of its factors. For let M and N be any two numbers, x and y their respective logarithms, and a the base of the system. Then, by definition (Art. 1), we have M= a, N=a". Multiplying equations, member by member, we have Therefore, MN a a" = a++". log (MXN)=x+y=log M+log N. 10. The logarithm of a QUOTIENT is equal to the logarithm of the dividend diminished by that of the divisor. For, by Art. 9, we have M= a, Na". Dividing the first equation by the second, member by member, we have 11. The logarithm of any POWER of a number is equal to the product of the logarithm of the number by the exponent of the power. For let m be any number, and take the equation (Art. 9) M=a*, then, raising both sides to the mth power, we have Therefore, Mm=(ax)=aTM. log (Mm)=x m = (log M) × m. 12. The logarithm of the ROOT of any number is equal to the logarithm of the number divided by the index of the root. For, let n be any number, and take the equation (Art. 9) M= a*, then, extracting the nth root of both sides, we have |