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Let c be the base in the system whose modulus is km; then since y = 1 when x = b, and #y = logen a = log on c = I, when x = e, * = expen k = c^. and c= expo, ; = b /*.* Hence: If the modulus be changed from m to km, the corresponding base is changed from 5 to b%. Again, if in the equation logo, a = m. ln c, č be substituted for x, the value of m is obtained in the form m = 1 /ln b. Hence in terms of the base of a system the logarithm is logn w = ln & /ln b, and if to x the value e be given, m is obtained in the form m = logne; whence passing to the corresponding inverse relation:
that is: The exponential of any quantity with respect to itself as a modulus is equal to natural base.
25. The Law of Involution. By virtue of the fundamental principle of the last article—that to multiply the modulus multiplies the logarithm by the same amount— we have in general expen 2 = expm 2 / 4 and expm & 2 = expm &2. Hence expm / & = expm / K = expm & A = expm/hk I,
* For an amplification of this proof see Appendix, page 139.
or in other terms, since changing m into m / h, or m / 4, changes b into b”, or b%,
Ahk - (bh) K = (#) h, which expresses the law of involution. Obviously h, or k,
or both, may be replaced by their reciprocals in this formula and the law, more completely stated, is
(5%) A = (6%) ×h. Evolution. If k be an integer, the process indicated by
bo” is called evolution; and when k = 2, it is usually expressed by the notation 1/ b.
26. The Law of Metathesis. Let z = b%; then by the law of involution
2* = (5%) * = book; whence, passing to the corresponding inverse relations,
which expresses the law of interchange of exponents with coefficients in logarithms, the law of metathesis.
27. The Law of Indices.* The law of indices, or of addition of exponents, follows very simply from the definition of the exponential, thus: In the construction of Art. 23, and by virtue of Definition (iii) of that Article
* For an alternative proof see Appendix, pages 139-14t.
29. Infinite Values of a Logarithm. If, in the construction of Art. 23, A and p become larger than any previously assigned arbitrarily large value, while their ratio m (that is, the modulus) remains unchanged, P and Q are transported instantly to an indefinitely great distance, and OP, OQ become simultaneously larger than any assignable magnitude. It is customary to express this fact in brief by writing
b” = 0, logn oc = 0: ;
though to suppose these values actually attained would require both A and p to become actually infinite. This supposition will be justifiable whenever we find it legitimate, under the given conditions, to assign to the indeterminate form p / A = 00 / Co a determinate value m.*
In like manner, since from bo = Co we may infer b -* = 1 / 2 = O, the equations
& To = O, logo, O = — Co
are employed as conventional renderings of the fact, that when P and Q are moving to the left, P passing from / towards O and Q negatively away from O, v, in the equations & = &o, y = logo, ac, remains positive and approaches o, while y is negative and approaches — Co.
30. Indeterminate Exponential Forms. When z. X logo, u becomes either + O X co, or + Co X O, it is indeterminate (Art. I I). Now logo, u is O if u = I, is + Co if u = -- oc, and is — oc if u = o (Art. 29). Hence v X logo, u will assume an indeterminate form under the following conditions:
*This form of statement must be regarded as conventional. Strictly speaking we cannot assign a value to an indeterminate form. When the quotient a sy, in approaching the indeterminate form, remains equal to, or tends to assume, a
definitive value, we substitute this value for the quotient and call it a limit. In conventional language the indeterminate form is then said to be evaluated.
But if v × logo, u, or logo, u” is indeterminate, so is u", and therefore the forms 1 °, oo", O" are indeterminate. Whenever one of these forms presents itself, we write y = u” and, operating with ln, examine the form ln y = v × 1n u.
If then ln y can be determined, y can be found through the equation y F eln 3.
VIII. SYNOPSIS OF LAws OF ALGEBRAIC OPERATION.”
31. Law of Signs:
(i). The concurrence of like signs gives the direct sign, + or X. Thus:
even number of negative factors, in a product or quotient, gives a positive result. Thus:
(iv). The concurrence of an odd number of negative factors, in a product or quotient, gives a negative result.
* Cf. Chrystal: Algebra, vol. I, pp. 20–22.