21. x3 in (1-x2 + x3 − x3)1. 22. x in (1+ ax + bxo) ̄. 23. x3 in (1+ax +α ̧x2 + α ̧23 + ......)”. 26. Find the coefficient of ab*c3 in (a+b+c+d)o. 27. Find the coefficient of ab3c3d1 in (a − b + c - d)1o. 28. Write down those terms in the expansion of (a+b+c)" which involve powers of b and c as high as the third power inclusive. 29. Write down all the terms in the expansion of which contain d"-3. (a+b+c+d)" Find the greatest coefficient in the expansion of 31. The greatest coefficient in the expansion of where 2 by m. is the quotient, and r the remainder when n is divided 32. Shew that the coefficient of x2+1 in the expansion of ...... 36. In the expansion of (1 + x + 203 + +x")", where n is a positive integer, shew that (1) the coefficients of the terms equi T. A. 20 distant from the beginning and the end are equal; (2) the coefficient of the middle term, or of the two middle terms, according as n is even or odd, is greater than any other coefficient; (3) the coefficients continually increase from the first up to the greatest. ... be the coefficients in order of the expansion of (1 + x + x2 + ......+x")", prove that 531. Suppose a* = n, then x is called the logarithm of n to the base a; thus the logarithm of a number to a given base is the index of the power to which the base must be raised to be equal to the number. The logarithm of n to the base a is written log. logan = x expresses the same relation as a = n. n; thus 532. For example 3*=81; thus 4 is the logarithm of 81 to the base 3. = ...... If we wish to find the logarithms of the numbers 1, 2, 3, to a given base 10, for example, we have to solve a series of equations 101, 10* 2, 10 = 3, We shall see in the next chapter that this can be done approximately, that is, for example, although we cannot find such a value of x as will make 10* = 2 exactly, yet we can find such a value of x as will make 10 differ from 2 by as small a quantity as we please. We shall now prove some of the properties of logarithms. 533. The logarithm of 1 is 0 whatever the base may be. For a=1 when x = 0. 534. The logarithm of the base itself is unity. For aa when x= = 1. 535. The logarithm of a product is equal to the sum of the logarithms of its factors. For let therefore, therefore, therefore, x = log1m, y = log. n; m = a*, n = a3 ; mn = a*a3 = a*+y; log. mn = x + y = log. m + log,n. 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*", log. (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 ʼn may be found by multiplying the logarithm of the number to the base a by We may notice that log, a × log,b=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. 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. For log, 10" × N = log11 N + log1, 10′′ = log1。 N + n, log10 10 10 N · log1。 N — log1 10′′ = log10 10 - N. 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 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, and of log, (3). 4. Find log, 3125. 5. Give the characteristic of log,, 1230, and of log, 0123. 6. Given log 2 = 301030 and log 3 = 477121, find the logarithms of 05 and of 5.4. of ⚫006. Given log 2 and log 3 (see question 6), find the logarithm 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 log,, 2301030 and log,, 7·845098, find logo 98 and the logarithm of 4 343 to the base 1000. |