student should notice that for all positive integral values of n, (1 + x)" = ƒ(n), as, (1 + x)3 = f(3); and that it remains to be proved that (1 + x)" = f(n), when n is a fraction or negative; as, for example, that (1 + x) = ƒ(}). We will assume that the product f(m) ×f(n) is a convergent series, when the two series are convergent. The proof of this principle is beyond the scope of this book. We then have (1) therefore f(m) ׃(n) = f(m + n), for all rational values of m and n. Then f(m)×f(n) ׃(p)= f(m+n)׃(p)= f(m+n+p). In general, f(m) ×f(n) × ƒ(p) × ··· × ƒ(r) = f(m + n + p + ··· +r), (2) ... for all rational values of m, n, p, ..., r. ... 11. Fractional Exponents. Let - и m = n = p = ··· = r = wherein u and v are positive integers. Taking v factors, we now have Now, since u is a positive integer, (1 + x)" = f(u). 12. Negative Exponents, Integral or Fractional. In (1), Art. 10, let m = n. We then have ƒ(-n) × ƒ(n) = f(n−n) = f(0) = 1, Since n is a positive integer or fraction, (1 + x)" = ƒ (n), In a similar way it can be shown that, when a is numerically Notice that when n is a fraction or negative, formula (3) or (4) must be used according as a is numerically greater or less than b. 1 14. Ex. Expand to four terms. 3 (a− 4 b2) If we assume a > 4 b2, we have, by (3), Art. 13, If a < 4 b2, we should have used (4), Art. 13. Any particular term can be written as in Ch. XXII., Art. 9. 15. Extraction of Roots of Numbers. - Ex. Find √17 to four Therefore √17 = 4.1231, to four decimal places. 16. 4th term of (1 − 2 x)3. 17. 6th term of (1 + a2b ̄3)−3. 18. 5th term of (x — x ̄1y2) ̄‡. Find to four places of decimals the values of: 22. √5. 23. √27. 24. $/35. 25. 700. 27. Find the term in (3 x3 — x2y) containing 2. 28. Find the term in (a+1)* containing a-u. 26. 258. CHAPTER XXVIII. LOGARITHMS. 1. A value of x can always be found to satisfy an equation of the form 10 = n, wherein n is any real positive number. E.g., when n = = 10, x=1, when n = 100, x2, when n = 1000, x 3, etc. = The proof of this principle is beyond the scope of this book. When n is not an integral power of 10, the value of x is irrational, and can be expressed only approximately. Thus, when = 24, the corresponding value of x has been found to be 1.38021..., to five decimal places; or n = 101.38021... = 24. A value of a is called the logarithm of the corresponding value of n, and 10 is called the base. In general, a value of a which satisfies the equation b is called the logarithm of n to the base b. = Ng E.g., since 28, 3 is the logarithm of 8 to the base 2; since 102 = 100, 2 is the logarithm of 100 to the base 10. The Logarithm of a given number n to a given base b is, therefore, the exponent of the power to which the base b must be raised to produce the number n. 2. The relation b = a is also written x = log, a, read x is the logarithm of a to the base b. Thus, are equivalent ways of expressing one and the same relation. |