SECTION XXI. LOGARITHMS. (335.) In a system of logarithms, all numbers are considered as the powers of some one number, arbitrarily assumed, which is called the base of the system; and the exponent of that power of the base which is equal to any given number is called the logarithm of that number. Thus, if a be the base of a system of logarithms, and a2=N, then 2 is the logarithm of N; that is, 2 is the exponent of the power to which the base (a) must be raised to equal N. If a3=N', then 3 is the logarithm of N' for the same reason; and if a=N", then x is called the logarithm of N" in the system whose base is a. The base of the common system of logarithms (called, from their inventor, Briggs' Logarithms) is the number 10. Hence in this system all numbers are to be regarded as powers of 10. Thus, since From this it appears that, in Briggs' system, the logarithm of every number between 1 and 10 is some number between 0 and 1, i. e., is a proper fraction. The logarithm of every number between 10 and 100 is some number between 1 and 2, i. e., is 1 plus a fraction. The logarithm of every number between 100 and 1000 is some number between 2 and 3, i. e., is 2 plus a fraction, and so on. (336.) The preceding principles may be extended to fractions by means of negative exponents. Thus, 10-1 or i =0.1; therefore, 1 is the logarithm of .1 in Briggs' system. .0001. Hence it appears that the logarithm of every number between 1 and .1 is some number between 0 and 1, or may be represented by 1 plus a fraction; the logarithm of every number between .1 and .01 is some number between 1 and -2, or may be represented by -2 plus a fraction; the logarithm of every number between .01 and .001 is some number between 2 and 3, or is equal to -3 plus a fraction, and so on. (337.) The logarithms of most numbers, therefore, consist of an integer and a fraction. The integral part is called the characteristic, and may always be known from the following RULE. The characteristic of the logarithm of any number greater than unity, is one less than the number of integral figures in the given number. Thus the logarithm of 297 is 2 plus a fraction; that is, the characteristic of the logarithm of 297 is 2, which is one less than the number of integral figures. The characteristic of the logarithm of 5673 is 3; of 73254 is 4, &c. The characteristic of the logarithm of a decimal fraction is a negative number, and is equal to the number of places by which its first significant figure is removed from the place of units. Thus the logarithm of .0046 is 3 plus a fraction; that is, the characteristic of the logarithm is -3, the first significant figure, 4, being removed three places from units. In a series of fractions continually decreasing, the negative logarithms continually increase. Hence, if the fraction is infinitely small, its logarithm will be infinitely great; that is, in Briggs' system, the logarithm of zero is infinite and negative. U GENERAL PROPERTIES OF LOGARITHMS. (338.) Let N and N' be any two numbers, x and x' their re- ▾ spective logarithms, and a the base of the system. Then, by the definition, Art. 335, Multiplying together equations (1) and (2), we obtain NN'=aa Therefore, according to the definition of logarithms, x+x' ıs the logarithm of NN', since x+x' is the exponent of that power of the base a which is equal to NN'; hence PROPERTY I. The logarithm of the product of two or more factors is equal to the sum of the logarithms of those factors. Hence we see that if it is required to multiply two or more numbers by each other, we have only to add their logarithms; the sum will be the logarithm of their product. We must then look in the Table for the number answering to that logarithm, in order to obtain the required product. EXAMPLES. Ex. 1. Find the product of 8 and 9 by means of logarithms. On page 318, the logarithm of 8 is given 0.903090 0.954243 1.857333, which, according to the same Table, is seen to be the logarithm of 72. Ex. 2. Find the continued product of 2, 5, and 14 by means of logarithms. Ex. 3. Find the continued product of 1, 2, 3, 4, and 5 by means of logarithms. (339.) If, instead of multiplying, we divide equation (1) by equation (2), we shall obtain Therefore, according to the definition, x-x' is the logarithm N of since x-x' is the exponent of that power of the base a N" The logarithm of a fraction, or of the quotient of one number divided by another, is equal to the logarithm of the numerator, minus the logarithm of the denominator. Hence we see that if we wish to divide one number by another, we have only to subtract the logarithm of the divisor from that of the dividend; the difference will be the logarithm of their quotient. EXAMPLES. Ex. 1. It is required to divide 108 by 12 by means of logarithms. which is the logarithm corresponding to the number 9. Ex. 2. Divide 133 by 7 by means of logarithms. The preceding examples are designed to illustrate the properties of logarithms. In order to exhibit fully their utility in computation, it would be necessary to employ larger numbers; but that would require a more extensive Table than the one given on page 318. (340.) Logarithms are attended with still greater advantages in the involution of powers and in the extraction of roots. For if we raise both members of equation (1) to the mth power, we obtain ⚫ Therefore, according to the definition, ma is the logarithm of N", since mx is the exponent of that power of the base which is equal to N"; hence PROPERTY III. The logarithm of any power of a number is equal to the logarithm of that number multiplied by the exponent of the power. EXAMPLES. Ex. 1. Find the third power of 4 by means of logarithms. Ex. 2. Find the fourth power of 3 by means of logarithms. Ex. 3. Find the seventh power of 2 by means of logarithms. Ex. 4. Find the third power of 5 by means of logarithms. (341.) Also, if we extract the mth root of both members of equation (1), we shall obtain The logarithm of any root of a number is equal to the loga. rithm of that number divided by the index of the root. EXAMPLES. Ex. 1. Find the square root of 81 by means of logarithms The logarithm of 81 is Divided by 2 The quotient is which is the logarithm of 9. 1.908485 .954243, |