the corresponding exponents, we find 8192 is the 13th power of 2, and 32 is the 5th power. Looking for 8 in the column of exponents, and for its corresponding number, we find 256 for the 8th power of 2, or the quotient of 8192 by 32. Divide 32768 by 512. The exponents corresponding to these numbers in the table are 15 and 9. 15—9=6. In the column of exponents, 6 corresponds to 64, which is the true quotient of 32768 by 512 What is the 3d power of 32 ? The exponent corresponding to 32 is 5. Now to find the 3d power of a we should multiply the exponent by 3, thus 25X3 = a's. So the third power of 25 is 25x3 = 25. Against 15 in the column of exponents we find 32768 for the 15th power of 2. Therefore the 3d power of 32 is 32768. What is the ad power of 128 ? The exponent corresponding to this number is 7. 7 X2= 14. The number corresponding to the exponent 14 is 16384, which is the second power of 128. What is the 3d root of 4096 ? 11 The number corresponding to the exponent 4 is 16, which is the 3d root of 4096. What is the fourth root of 65,536 ? The exponent corresponding to this number is 16, which divided by 4 gives for the exponent of the root 4, the number corresponding to which is 16. The answer is 16. . Examples. 1. Multiply 512 by 256. 2. Multiply 8192 by 128. 3. Multiply 2048 by 256. · 4. Divide 262,144 by 128. 5. Divide 1,048,576 by 512. 6. Divide 524,288 by 131,072. 7. What is the 2d power of 1024 ? 8. What is the 3d power of 64? 9. What is the 5th power of 16? 10. What is the 2nd root of 262,144 ? 11. What is the 3d root of 262,144 ? 12. What is the 4th root of 1,048,576 ? 13. What is the 5th root of 1,048,576 ? 14. What is the 6th root of 262,144 ? The operations of multiplication, division, and the extraction of roots are very easy by means of this table. This table however contains but very few numbers. But an exponent of 2 may be found for all numbers from 1 as high as we please. For 2 = 2, and 22 = 4. Hence the exponent of 2 answering to the number 3 will be between 1 and 2; that is, 1 and a fraction. So the exponents answering to 5, 6, and 7, will be 2 and a fraction, &c. XLIX. A table may also be made of the powers of 3, or of 4, or any other number except 1, which shall have the same properties. Exponents might be found answering to every number from 1 upwards. 3° = 1, 3 = 3, 39 = 9, 38 = 27, &c. The column of powers will always consist of the numbers 1, 2, 3, &c. but the column of exponents will be different according as the numbers are considered powers of a different number. 1 The formula a? = y will apply to every table of this kind. If any number except 1 be put in the place of a, and y be made successively 1, 2, 3, 4, a suitable value may be found for X, which shall answer the conditions. If a be made 1, y will always be 1, whatever value be given to x; for all powers, as well as all roots of 1, are 1. But if any number greater than 1 be put in the place of a, y may equal any number whatever, by giving x a suitable value. Giving a value to a then, we begin and make y successively 1, 2, 3, 4, &c. and these numbers will form the first column or columns of powers in the table. Then we find the values of x corresponding to these values of y, and write them in the second column against the values of y, and these form the column of exponents. These exponents are called logarithms. The first column is usually called the column of numbers, and the second, the column of logarithms. The number put in the place of a, is called the base of the table. Whatever number is made base at first, must be continued through the table. Observe that a = 1; therefore whatever base be used, the logarithm of 1 is zero. And 1 will be the logarithm of the base, for a' = d. The most convenient number for the base, and the one generally used in the tables, is 10. 10o = 1, 10 = 10, 10% = 100, 10% = 1000, 10* = 10000, 10% = 100000, 10% = 1000000, &c. Now to find the logarithm of 2, 3, 4, &c. For all numbers between 1 and 10, x must be a fraction, because 10o = 1 and 10% = 10. As the process for finding the value of z in this equation is long and rather too difficult for young learners, we will suppose it already found. 30103 4_771 2 1 3 Hence 10 Tūūūūū = 2 very nearly. To understand this, we must suppose 10 raised to the 30103d power, and then the 100000th root of it taken, and this will differ very little from 2. The number .30103 is the logarithm of 2. The fractional part of logarithms is always expressed in decimals. Having the logarithm of 2, we may find the logarithm of 4 by doubling it, for 2 = 4. That of 8 = 23 is found by tripling it, and so on. The logarithm of 4 is .30103 X 2 = .60206. = 3 very nearly. Hence the logarithm of 3 is .4771213. Since 2 X 3 = 6, the logarithm of 6 is found by adding the logarithm of 2 and 3 together. .30103 +.4771213=.7781513 = logarithm of 6. Since 3 = 9, the logarithm of 9 is found by multiplying that of 3 by 2. With the logarithms of 2 and 3 the logarithms of all the powers of each, and of all the multiples of the two may be found. The logarithm of 5 may be found by subtracting that of 2 from that of 10, since 5 = 1. The logarithm of 10 is 1. 1–.30103 = .69897 = log. of 5. Now all the logarithms of all the multiples of 2, 3, 5, and 10 may be found. Hence it appears that it is necessary to find the logarithms of the prime numbers, or such as have no divisor except unity, by trial ; and then the logarithms of all the compound numbers may be found from them. The decimal parts of the logarithms of 20, 30, &c. are the same as those of 2, 3, 4, &c. For, since the logarithm of 10 is that of 100,2 ; that of 1000, 3, &c., it is evident that add 1; ing these logarithms to the logarithms of any other numbers, will not alter the decimal part. Hence 1 added to the logarithm of 2 forms that of 20, and 2 added to the logarithm of 2 forms that of 200, &c. Log. 2 = .30103, log. 20 = 1.30103, log. 200 = 2.30103, log. 2000 = 3.30103. The logarithm of 25 is 1.39794 ; that of 250 = 25 X 10 is 1 + 1.39794 = 2.39794; that of 2500 = 25 X 100 is 2 + 1.39794 = 3.39794. The logarithms of all numbers below 10 are fractions, those of all the numbers between 10 and 100 are 1 and a fraction ; those of all numbers between 100 and 1000 are 2 and a fraction; those of all numbers between 1000 and 10000 are 3 and a fraction. That is, the whole number which precedes the fraction in the logarithm is always equal to the number of figures in the number less one. This whole number is called the index or characteristic of the logarithm. Thus in the logarithm 2.3576423, the figure 2 is the characteristic showing that it is the logarithm of a number consisting of three figures or between 100 and 1000. As the characteristic may always be known by the number, and the number of figures in a number may be known by the characteristic, it is usual to omit the characteristic in the table, to save the room. It is useful to omit it too, because the same fractional part, with different characteristics, forms the logarithms of several different numbers. The logarithm of 37 is 1.568202. 37 = 3.7 = = 10.568202. 10 10 The logarithm of 3.7 is .568202, which is the same as that of 37, with the exception of the index. 3762 103-575419 102 101.568202 = 101.575419. |