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238. The above laws may be thus expressed :
III. Power-raising of a power is performed by multiplying indices.
IV. Root-taking of a power is performed by dividing indices.
Extension of Index Notation. 239. Each of the two inverse laws, viz., II. and IV., are proved under a condition. This condition is equivalent to the statement that the index on the right-hand side shall be positive (Law II.) and integral (Law IV.). Otherwise the right-hand side would be meaningless, for we have only defined the symbol 2."", when m is a positive integer.
It is convenient, however, to be able to use these formulæ for any case whatever. To do this we have only to interpret negative or fractional indices so that Laws II. and IV. shall hold universally. That is, for all positive integral values of m and n,
- shall equal .cm = cc", according to Law II., and amin shall equal »/(), according to Law IV.
In the formula of Law II., then
2cm-m must equal 2cm = 2c",
(1). Secondly, let m = 0, then 200-n must equal 2° = 2c",
1 i.e. x-n must equal
Thirdly, in the formula of Law IV., let m be not divisible by
240. The interpretations (1), (2), (3) of the last article, which are of the greatest importance, thus enable us to use the four index Laws for all positive integral values of m and n. It only remains to show that, with the interpretation of negative and fractional indices given in the last article, the same laws hold for all values of m and n whatever. This may be shown by use of the preceding articles. Thus, e.g.
x xc +å
Similarly xå = bd/(sobe),
= od' (pad+bc) (Law I.)
To prove acó x xå mota
Example 1. Write down a series of powers of x beginning with 23, in which each power is derived from the preceding by dividing by x.
co 1 -21;
Thus we see that negative indices simply carry out the rule that to divide by x is equivalent to subtracting 1 from the index of x. Example 2. Interpret x}, x3, x*.
(x2)2 must equal x£x, i.e. x1, i.e. x.
are two different ways of stating the same relation between x, m, and P. In the first, P stands by itself; in the second, x stands by itself. There is a third way of expressing this same relation, in which m stands by itself. Thus
If xm = P, m is called the logarithm of P to the base x; and we write m = log, P (the base being written below).
Der. A logarithm of a given number to any base is the index of that power of the base, which is equal to the given number.
It should be remembered, then, that a logarithm is simply an index made to stand by itself. Hence the laws of logarithms are simply the laws of indices expressed in different notation. The equations
2cm = P and m= log2 P have the same meaning: and either one of them may be substituted for the other. The student should examine the truth of the statement
2108, P = P.
Example 1. What is the logarithm of 8 to the base 2 ?
This means to what power must 2 be raised to give 8. Now 23=8; .: 3 is the logarithm of 8 to the base 2; or 3=log, 8.
Example 2. Write down the values of logz 81, log10 10000, loge 64. Since
.: 4=logz 81. Since 104 = 10000,
.: 4=log10 10000. Since 43=64,
.: 3=log, 64.
242. In using the relation 2cm = P or m = logą P, we shall assume for the present that x and P (i.e. the base and power) are both positive, and that x is not equal to 1.
243. To show that there is only one positive or negative* value of the logarithm of a positive power to a positive base (not equal to 1).
If possible let m and n be two different logarithms of P to the base x: so that
P=2cm = 2".
Now m and n being positive or negative, m – n must be positive, negative, or zero.
But, if x is not equal to 1, no positive or negative power of x can be equal to 1. For
If x > 1, positive powers of x are > l; negative powers are <l.
.. m-n must be zero. .. m=n.
i.e. the two logarithms supposed to be different are not different.
Cor. In the same manner it may be shown that, if x > 1, the logarithms of numbers to the base x increase as the numbers increase : but, if x < 1, the logarithms of numbers to the base x decrease as the numbers increase.
244. In order to prove any formula involving logarithms, the student has simply to translate from the logarithmic language with which he is unfamiliar into the index language with which he is more familiar.
Thus he has only to remember that the two equations
acm = P and m= log. P are precisely equivalent statements in the two different languages of indices and of logarithms.
h * Whether there are logarithms which are neither positive or negative is not here discussed.
The Logarithmic Laws involving the same base. 246. Law I. log, (m xn) = logc m + logen.
Law II. logx (m + n) = logxm - logan.
Law IV. loge (m/m) = (logæ m) = n.
247. To prove
logx (m = n) = log2 m - logen. Law II.
b; this means rcb
logx (m n) = a + b, i.e. logæ m + logx n.