534. The ogarithm of the base itself is unity. For at a when x = 1. 535. The logarithm of a product is equal to the sum of the logarithms of its factors. 536. The logarithm of a quotient is equal to the logarithm of the dividend diminished by the logarithm of the divisor. 537. The logarithm of any power, integral or fractional, of a number is equal to the product of the logarithm of the number by the index of the power. For let therefore m = a*; therefore m" = (a*)* = a*, loga (m") = xr = r log。m. 538. To find the relation between the logarithms of the same number to different bases. Hence the logarithm of a number to the base b may be found by multiplying the logarithm of the number to the base a by We may notice that log, a × log1b = 1. 539. In practical calculations the only base that is used is 10; logarithms to the base 10 are called common logarithms. We will point out in the next two articles some peculiarities which constitute the advantage of the base 10. We shall require the following definition; the integral part of any logarithm is called the characteristic, and the decimal part the mantissa. 540. In the common system of logarithms, if the logarithm of any number be known we can immediately determine the logarithm of the product or quotient of that number by any power of 10. That is, if we know the logarithm of any number we can determine the logarithm of any number which has the same figures, but differs merely by the position of the decimal point. 541. In the common system of logarithms the characteristic of the logarithm of any number can be determined by inspection. For suppose the number to be greater than unity and to lie between 10" and 10"+1; then its logarithm must be greater than n and less than n+1; hence the characteristic of the logarithm is n. Next suppose the number to be less than unity, and to lie 1 between and 10" 1 that is, between 10" and 10"+1); then its logarithm will be some negative quantity between -n and -(n+1); hence if we agree that the mantissa shall always be positive, the characteristic will be − (n + 1). Further information on the practical use of logarithms will be found in works on Trigonometry and in the introductions to Tables of Logarithms. EXAMPLES OF LOGARITHMS. 1. What is the logarithm of 144 to the base 2√3? 2. What is the characteristic of the logarithm of 7 to the base 2 ? 3. Find the characteristic of log, 5. 4. Find log, 3125. 5. Give the characteristic of log,, 1230, and of log1 0123. 6. Given log 2=301030 and log 3 = 477121, find the logarithms of '05 and of 5.4. 7. Given log 2 and log 3 (see Ex. 6), find the logarithm of '006. 8. Given log 2 and log 3, find the logarithms of 36, 27, and 16. 9. Given log 648 = 2.81157501, log 864 = 2·93651374, find log 3 and log 5. 10. Given log 2, find log (1.25). 11. Given log 2, find log 0025. 12. Given log 2, find log (0125). 13. Given log 2 and log 3, find log 1080 and log (0025)*. 14. Having given log1, 2301030 and log,, 7845098, find 10 log, 98 and the logarithm of 15. Find the number of digits in 264, having given log 2. 16. Given log 2, and log 5-7434917591760, find the fifth root of 0625. 17. If P be the number of the integers whose logarithms have the characteristic p, and Q the number of the integers the logarithms of whose reciprocals have the characteristic -q, shew that 18. If y=e 1 log P-log Q=p-q+1. 1 1 1-log% 1-log and z = e' 19. If a, b, c be in G. P., then log, n, log, n, log, n are in H. P. 20. If the number of persons born in any year be th of 45 the whole population at the commencement of the year, and the number of those who die th of it, find in how many years 1 60 population will be doubled; having given log 2301030, log 1802-255272, log 181 = 2.257679. the XXXIX. EXPONENTIAL AND LOGARITHMIC SERIES. 542. To expand a" in a series of ascending powers of x; that is, to expand a number in a series of ascending powers of its logarithm to a given base. a* = {1 + (a− 1)}" = 1 + x (a− 1) + x (x − 1) (a-1) · 1 + x {a − 1 − 1 ( a − 1 )2 + § (a − 1 )3 − 1 ( a − 1)* + ...... } + terms involving x2, x3, &c. This shews that a can be expanded in a series beginning with 1 and proceeding in ascending powers of x; we may therefore suppose that are quantities which do not depend on x, and which therefore remain unchanged however x may be changed; also c1 = a-1-(a−1)2 + § (a − 1)3 — — (a − 1)* + ...... ...... while C2, C39 are at present unknown; we proceed to find their values. Changing x into x + y we have a2+y=1+c, (x + y) + c ̧ (x + y)2 + c ̧ (x + y)3 + ; 3 ....}. but a2+y = a*a3 = a3 {1 + c‚x + c2x2 + c2x3 + Since the two expressions for a*** are identically equal, we may assume that the coefficients of x in the two expressions are equal, thus c1 +2c2y+3c ̧y2 + 4c ̧y3 + = c1ay In this identity we may assume that the coefficients of the corresponding powers of y are equal; thus Since this result is true for all values of x, take x such that this series is usually denoted by e; thus ao1e, therefore a = ec1 |