4. Find the value of log, 8, log2.5, log, 243, logs .04, log10 1000, log10.001. 5. Find the value of log, a log, V62 logs 2, log27 3, log100 10. 3 6. Prove that log (VZ × V7 ÷ V9) = log 2 + log 7 — log 3. 121. That system of logarithms whose base is 10 is called the common system of logarithms. In speaking of logarithms hereafter, common logarithms are referred to unless the contrary is expressly stated. We shall assume that an index of 10 can be found such that 10 affected with this index is practically equivalent to any number. The indices of these powers of 10, i.e. the common logarithms, are in general incommensurable numbers. Now, the greater the index with which 10 is affected, the greater will be the value of the equivalent expression; and the less the index, the less will be the numerical value of the expression. Hence, if one number be less than another, the logarithm of the first will be less than the logarithm of the second. But the student should notice that logarithms (or indices) are not proportional to the corresponding numbers. find EXAMPLE. 1000 is less than 10000; and the logarithm to base 10 of the first is 3 and of the second is 4. But 1000, 10000, 3, 4 are not in proportion. 122. PROPOSITION. If two numbers expressed in the decimal notation have the same digits arranged in the same order (so that they differ only in the position of the decimal point), their logarithms to the base 10 differ only by an integer. The decimal point in a number is moved by multiplying or dividing the number by some power of 10. Let the numbers be m and n; then m = n× 10* when k is a whole number (positive or negative); then log m = log (n × 10*) = log n + log 10* = log n+k. (Art. 120.) That is, log m and logn differ by an integer. Q.E.D. EXAMPLE 1. log 1679.2 = log {(1.6792) × 1033 = log 1.6792 + log 103 log 1.6792 +3. Here EXAMPLE 2. Given that log 1.7692.247776; log.0017692 = log (1.7692 × 10−3) = − 3 + .247776, log 176.92 = log (1.7692 × 102) = 2.247776. Hence, the logarithm of 1 is 0. The logarithm of any number greater than 1 is positive. The logarithm of any positive number less than 1 is negative. 124. Observe also that the logarithm of any number between 1 and 10 is a positive decimal fraction; that the logarithm of any number between 10 and 100, i.e. between 101 and 102, is 1+ a decimal fraction; EXAMPLE 2. Given that 3: = integral part of 320. We have that the logarithm of any number between 1000 and 10000, i.e. between 10 and 104, is 3+ a decimal fraction; and so on. 125. Observe also that the logarithm of any number between 1 and .1, .e. between 10° and 10-1, can be written in the form - 1+ a decimal fraction; that the logarithm of any number between .1 and .01, i.e. between 10- and 10-2, can be written in the form - 2+ a decimal fraction; and so on. EXAMPLE 1. How many digits are contained in the integral part of the number whose logarithm is 3.67192 ? 3 = 10.4771213, 320 =(10.47712) 20 The number is 103.67192 and this is greater than 103, i.e. greater than 1000, and it is less than 104, i.e. less than 10000. Therefore the number lies between 1000 and 10000, and therefore the integral part of it contains four figures. 10-4771213, find the number of the digits in the =109.54242 Therefore there are 10 digits in the integral part of 320; for it is greater than 109 and less than 1010. EXAMPLE 3. Suppose that the decimal part of the logarithm is to be kept positive, find the integral part of the logarithm of .0001234. This number is greater than .0001, i.e. than 10-4 and less than .001, i.e. than 10-8. Therefore its logarithm lies between - 3 and - 4+ a fraction; the integral part is therefore - 4. - - 4, and therefore it is For 126. From Art. 120-125 it is evident that the logarithm of any positive number may be written as an integer + a decimal fraction. The integral part of the logarithm is called the characteristic. The decimal part of the logarithm is called the mantissa. convenience, the mantissæ of common logarithms are always kept positive. In this way the mantissæ of the logarithms of numbers consisting of the same digits, arranged in the same order, are always the same (Art. 120); because removing the decimal place to the right or to the left is equivalent to multiplying the number by 10*, where k is a positive or negative integer, as the case may be. EXAMPLE. The mantissa of log 3.456 mantissa of log (345.6). The student cannot observe too carefully that the mantissa is always positive. The mantissæ have been calculated and arranged in convenient tables. See table I. 113 127. It is evident from Arts. 120–125, that the characteristic of a logarithm can be obtained by the following rule: RULE. The characteristic of the logarithm of a number greater than unity is one less than the number of figures in the integral part of the number. The characteristic of a number less than unity is negative, and (when the number is expressed as a decimal) is one more than the number of ciphers between the decimal point and the first significant figure to the right of the decimal point. When the characteristic is negative, as for example in the logarithm −3+.17609, the logarithm is abbreviated thus, 3.17609. Thus EXAMPLE 1. The characteristics of 36741, 36.741, .0036741, 3.6741, and .36741 are respectively 4, 1, 3, 0, and 1. and so on. EXAMPLE 2. Given that the mantissa of the logarithm of 36741 6515, we can at once write down the logarithm of any number whose digits are 36741. log 3674100 = 6.56515, log 36741 = 4.56515, log 367.41 = 2.56515, log .36741 = = 1.56515, log .00036741 = 4.56515, 128. We have said that logarithms are in general incommensurable numbers. Their values can, therefore, only be given approximately. If the value of any number is given to seven significant figures, then the error (i.e. the difference between the given value and the exact value of the number) is less than a millionth part of the number. EXAMPLE. 3.141592 is the value of correct to seven significant figures. The error is less than .000001; for is less than 3.141593, and greater than 3.141592. The ratio of .000001 to 3.141592 is equal to 1: 3141592. to is less than this; i.e. much less than the ratio of one to one million. 129. An actual measurement of any kind must be made with the greatest care, with the most accurate instruments, by the most skilful observers, if it is to attain to anything like the accuracy represented by 'seven significant figures'; and, indeed, the value of any quantity given correct to 'four significant figures' is exact for most practical purposes. The ratio of .000001 130. A five-place table of logarithms is placed at the end of the book. (See table I.) Page 1 of this table contains the logarithms, to five places of decimals, of all numbers from 1 to 100. Pages 2-16 contain the mantissæ, to five decimal places, of the logarithms all numbers from 100 to 10000. But all numbers from 0 to ∞ is one of these numbers multiplied by ten affected with either a positive or negative index, e.g. 46283264.628326 × 10°; .03986 = 3.986 × 10-2. Hence by prefixing to the mantissæ the proper characteristic (see Art. 126) we obtain the logarithm of any number, of not more than four significant figures, from 0 to ∞o. 131. To find the logarithm of a given number. (a) If the number contains not more than four significant figures. Find the mantissæ from the table corresponding to these four significant figures and prefix the proper characteristic. The result is the logarithm required. Referring to the table I., page 8, we find, at the intersection of the row headed 406 and the column headed 4, the number .60895. .. log 4064. = 3.60895. EXAMPLE 2. To find log.04064. This logarithm differs from the former as to the characteristic, which is 2. .. log .04064 = 2.60895. (b) To find the logarithm of a number of more than four digits. We shall assume that if the difference is small the difference between numbers is proportional to the difference of their logarithms. This proposition is proved in works on Algebra. If there are more than four digits in the number, we cannot obtain its logarithm directly from the table, but must interpolate. This is illustrated by the following example. To find the logarithm of 3456.4. This number lies between 3457 and 3456. Its logarithm therefore lies between log 3457 and log 3456. log 3457 - log 3456 = 3.53870 - 353857 = .00013. Since increasing 3456 by 1 increases its logarithm by .00013 increasing 3456 by .4 of 1 increases its logarithm by .4 of .00013 or by .000052. In forming such products as .4 x .00013, we retain only five decimal places. We increase the number occupying the fifth place by unity if the succeeding number is equal to or is greater than 5. We neglect the number occupying the sixth place if it is less than 5. 132. To find the number whose logarithm is given. The method of procedure is just the reverse of that of Art. 131, and will be illustrated by the following examples: |