1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576,, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 410, &c. 10°,101,102, 103, 104, 105, 106, 107, 108, 109, 1010, &c. the terms of the arithmetical series, or the indices of the geometrical progressions, are called the Logarithms of the corresponding terms of the geometrical series. The terms of the arithmetical series, with the corresponding terms of each geometrical series, form a System of Logarithms, and the ratio 2, 3, 4, &c....10, &c., is called the base of the system of logarithms. In the preceding series, it is observed that the numbers 1, 2, 3, 4, &c., which are the terms of the arithmetical series, or the indices of the geometrical series, correspond to different numbers, according as the base is 2, 3, 4, &c,; for instance, 5 corresponds to 32 when the base is 2, to 243 when the base is 3, to 1024 when the base is 4; then, the same number may be the logarithms of several quantities. Moreover, we notice the same number in different series; for instance, 64 corresponds to 6 when the base is two, to 3 when the base is 4; likewise, 1024 corresponds to 10 when the base is 2, to 5 when the base is 4; then, the same number may correspond to different logarithms. Hence, in order to ascertain, on one side, which logarithm corresponds to a given number; and on the other side, which number corresponds to a given logarithm, it is necessary to know the base of the system of logarithms. 378. If any two terms of the arithmetical series, or any two indices, be added up, as 2+5=7, and the corresponding terms of the geometrical series be multiplied by each other, viz., 4x 32128 (when the ratio is 2), this product corresponds to the sum; also 3+7=10, the corresponding terms (when the ratio is 4), are 64 × 16384=1048576. Hence, the product of two terms of the geometrical series is equal to the sum of the logarithms of the corresponding terms. 379. Let us divide two terms of the G.S. by one another; for instance, 19683 by 81 (when the ratio is 3), the quotient is 243, and subtract 4, logarithm of 81 from 9, logarithm of 19683, the difference is 5, which corresponds to 243, the quotient; also (when the ratio is 4) we have 65536÷1024-64, and 8-5 =3, which corresponds to 64. Hence, the quotient of two terms of the G.S. is equal to the difference of the logarithms of these terms. 380. If any term of the A.S., or if any index, as 4, be doubled, the result, 8, is the logarithm of 256 (when the ratio is 2), which is the square of 16 corresponding to 4, the given term; the result 8 is also the logarithm of 6561 (when the ratio is 3), which is the square of 81, corresponding to 4, the given term. If a term of the A.S., or if an index, as 3, be trebled, the result, 9, is the logarithm of 19683 (when the ratio is 3), which is the cube of 27, corresponding to 3. Hence, the power of any number of the G.S. is formed by multiplying its logarithms by its index. From this last observation it follows, that if any term of the A.S., or if any index, be divided by 2, 3, 4, &c., the 2nd, 3rd, 4th, &c., root of the corresponding term of the G.S. will be determined; for example, 8÷2=4, the logarithm of 81 (when the ratio is 3), which is the square root of 6561, corresponding to 8; likewise, 8÷4=2, the logarithm of 16 (when the ratio is 4), which is the fourth root of 65536, corresponding to 8. 381. These properties enable us to form an idea of the importance of logarithms in facilitating certain arithmetical operations. As the base of the system of logarithms in use the number 10 has been adopted, as presenting great advantages over every other; then : 0=log 10° or 1, 1=log 101 or 10, 2=log 102 or 100 3 log 103 or 1000, 4 log 10 or 10000, 5 log 10% or 100000 6 log 10 or 1000000, &c. It follows from this that the logarithms of the numbers 10, 100, 1000, 10000, &c., are alone whole numbers; the logarithm of a quantity between 1 and 10 will be between 0 and 1, or a fraction; that of a quantity between 10 and 100 will be between 1 and 2, or 1+a fraction; that of a quantity between 100 and 1000 will be between 2 and 3, or 2+a fraction, and so on. 382. Suppose. now, it were required to find the logarithm of 5. Since 5 is between 1 and 10, find a geometric mean x of 1 and 10 (§345), then x= √10=3.1622776; also an arithmetical mean X of 0 and 1 (§344); then X= or .5, which is evidently log 10 the log of 10, since log 10: 4. 2 Because 5 is between 3.162...and 10, find the geometric mean of 3.162...and 10, and the arithmetic mean of .5 and 1. Then x= 31.622766, or 5.623, and X=. is the logarithm of 5.623. 1.5 2 =.75. Therefore, .75 Because 5 is between 3.162 and 5.623...continuing the same process upon these and other numbers, we shall at last find two quantities differing as little as possible from 5; and there will not be any sensible error in taking 5 for one geometric mean, and the corresponding arithmetic mean is the logarithm of 5. Thus, by similar operations, we shall determine the logarithm of every quantity. We may remark that it would only be necessary to calculate the logarithms of prime numbers, for the logarithms of multiples can be found by their factors. Having shown a method for finding the logarithms of all numbers, the results form a Table of Logarithms. 383. A logarithm consists of two parts, one on the left of the decimal point, called the characteristic, and the other, on the right, called the mantissa. It must be observed, that the characteristic contains as many units, but one, as there are figures in the corresponding number. 384. We owe the invention of logarithms to John Napier, a Scotch nobleman, born in 1550; but to Briggs, professor of geometry at Oxford, we are indebted for many improvements: he published the first table of logarithms, in 1624. With regard to the directions for using the tables, every necessary information will be found in the introduction accompanying the Tables of Logarithms. We shall, therefore, suppose every student familiarized with the manner of finding the logarithms of any given number; and conversely, of finding the number corresponding to any given logarithms. 385. Here are some applications : Ex. 1. What is the product of 564 and 792? log of product=5.6500043 Ex. 2. Find the quotient of 37812 and 454. log 37812-4.5776296 log 454 2.6570559 ..log of quotient=1.9205737 = Ex. 3. Determine by logarithms the value of the expression 54.6 × 1.764 log (9) 15 = =1.259921. 15 15(log 6-log 5) = 15 x 0.07918125 log 1.1877187 = 15.407. Ex. 6. £24. 12s. 6d. (3.56)4. log £24. 12s. 6d.=log 24.625=1.3913762 log (3.56)=4 log 3.56 2.2058000 log of answer=3.5971762 .. answer=£3955.271=£3955. 58. 5 d. nearly. Ex. 7. What is the amount of £1210, left unpaid for 54 years 6 months, at 3 per cent. per annum, C.I.? .. log a=log 1210+54.5 × log 1.035=log 3.8970318. .. amount = £7889.18= £7889. 3s. 7d. Ex. 8. In how many years will a principal double itself, at 3 per cent. per annum, Č.Ì. ? Here 2 P= P(1.03)". .. 2=(1.03)". .. n log 1.03=log 2. log 2 0.3010300 log 1.03 0.0128372 386. EXERCISES. 1. Determine a mean proportional between 64.5 and .73. What will £1000 amount to in 64 years, at 4 per cent. per annum, C.I.? 3. In what timę, at C.I., reckoning 4 per cent. per annum, will £100 amount to £1000? 4. At what rate per cent. per annum will £400 amount to £1600. 12s. 6d., in 12 years, C.I.? 5. Find the value of the expression (564)7; also of |