SECTION LXXII. On Logarithms. 1. It is required to find the logarithm of As is 1 divided by 10, we may find its logarithm by subtracting the logarithm of 10 from the logarithm of 1. The logarithm of 10 is 1, and the logarithm of 1 is nothing. Hence the logarithm of is 0—1, or — 1. By subtracting the logarithm of 10 from the logarithm of 2,. we get the logarithm of. The logarithm of 2 is .30103, the logarithm of 10 is 1: Hence the logarithm of is - 1.30103. 3 2. What is the logarithm of ? ? ? ? 18? 1/%? 3. What is the logarithm of, or 1.1? 1.2? 1.3? 1.4? 6.7? In order to obtain the logarithm of any number consisting of a whole number and a decimal, find the logarithm of the whole number expressed by the digits, and subtract as many units from the characteristic as there are decimal figures in the number. It is evident that the characteristic of the logarithm of a decimal is a negative number. Now, in extracting roots, the logarithms of numbers must be divided. Suppose we wish to divide-1.30103 by 2, it is necessary to keep the characteristic a whole number. In order to do. this, we write -2+1 instead of 1, and we have 2+1.30103. Dividing this by 2, we have -1.65056, which is the logarithm of the square or second root of 2. What is the logarithm of of its second or square root? ? And what is the logarithm It will be observed that the characteristic of the logarithm of a fraction is negative; but the fractional part is positive. Hence, in adding or multiplying logarithms, if there be any to. carry from the fractional part, it will diminish the characteristic. In multiplying the last decimal, we have twice 8 = 16; and then we have twice the -1, which gives which is carried, makes the characteristic -2; but, the +1 - 1. An expedient is sometimes adopted to avoid using the negative characteristics. This is done by adding 10 to the characteristic, and after the operation is performed, subtracting it again. If the two logarithms, -1.80618 and 3.84509 are to be added together, we first increase each characteristic by 10, and we have 9.80618 17.65127 Subtracting the two tens or 20, which were added, we have This expedient however is, in general, of little practical utility. SECTION LXXIII, On Logarithms. A unit, divided by any number, is called the reciprocal of the number; is the reciprocal of 5, the reciprocal of 6, &c. Multiplying by the reciprocal of a number, is the same as dividing by the number itself. Thus, multiplying 10 by is the same as dividing 10 by 5, multiplying 12 by is the same as dividing 12 by 6. To find the logarithm of the reciprocal, we subtract the log arithm of a number from the logarithm of 1. 1. It is required to find the logarithm of . The logarithm of 1 is 0, the logarithm of 6 is 77815. Hence the logarithm of is 0.77815, or -.77815; but in order to keep the fractional part of the logarithm positive, we represent the logarithm of 1, which is 0, by 1+1, which is equivalent to 0. 1 4, 2. Find the logarithm of,,, 1, 4, 1, 1, 12, 13, 14, 16. The logarithm of the reciprocal of a number is called the Arithmetic complement of the logarithm of the number. Since dividing by a number gives the same result as multiplying by its reciprocal, subtracting the logarithm of a number must give the same result as adding the logarithm of its reciprocal; or, adding the logarithm of a number gives the same result as subtracting the logarithm of its reciprocal, Now, suppose we wish to find the result of the arithmetical operations indicated in the following expression, by logarithms, 8 X 10 16 we should, according to what has been explained, add the logarithm of 8 to the logarithm of 10, and subtract the logarithm of 16 from their sum; but, instead of subtracting the logarithm of 16, we may add the logarithm of, which will give the same result. We can perform multiplication by adding the logarithms of numbers, and division, by subtracting the logarithms of numbers; but, by using the arithmetic complements of the divi sors, we can perform both multiplication and division by addition. 3. Find the result of the arithmetical operations, indicated in the following expressions, by using logarithms and their arithmetic complements. 4. Find the value of x in the following equations: Taking the logarithm of both numbers of this equation, we have Log. of 2 x x = log. of 64 log. 64 x= log. 2 Dividing the logarithm of 64 by the logarithm of 2, will give x. Thus 30103) 1.80618 (6 =X 1.80618 How can you prove that 6 is the proper value of x? 5. Find the values of x in the following: 6. How would you find the logarithm of x in the following equation, supposing you knew the values of the letters in the second member of the equation? a ph x= C Ans. 1. Find the logarithm of p in the table, and multiply it by n. 2nd. Find the logarithm of a in the table, and add it to the product. 3rd. Find the logarithm of c in the table, and subtract it from the sum. The remainder will be the logarithm of x. 7. How do you find the number x, after you have found its logarithm? Ans. Look in the table until you find the logarithm, and in the opposite column you will find the number to which it belongs, and it is the value of x required. 8. How do you find the values of x in the following equations, supposing you know the values of the letters in the second members? Principles Respecting the Use of Letters. 1. What will represent the sum of four numbers, represented respectively, by the letters a, b, c, and d? Ans. a+b+c+d. |