7. In the proofs of the preceding principles, the base was assumed to be greater than 1. Similar reasoning will, however, apply when the base is less than 1 and positive.

For, if b<1 and positive, it can be shown that the irrational power is defined by the relation

Solution

m+1

b n <bi<bn.

of the Equation 6* = a.

8. (i.) First, let b>1. The powers

..., b-2, b-1, bo, b1, b2, ...,

increase toward the right beyond any positive number, however great, as the exponents increase without limit. For, by Ch. XVII., Art. 15,

(1+d)">1+ nd,

But

wherein n and d are positive. We can make 1 + nd greater than any assigned number, however great, by increasing n without limit. 1+d represents any number greater than 1.

The same series decreases toward the left below any assigned positive number, however small, as the absolute values of the exponents increase without limit, by Ch. XXVII., §.3, Art. 8. For b-"=

b

and << 1. Then a will either be equal to one of these powers or lie between two consecutive powers. In the former case, x is equal to a positive or a negative integer. In the latter case, x lies between two consecutive numbers of the series

2,

− 1, 0, +1, + 2,

....

Let b and b+1 be the two powers between which a is found to lie; i.e., bk < a <b1⁄4+1, wherein k is 0, or any positive or negative integer. Then x lies between k and k + 1; or, k < x <k + 1.

The interval between k + 1 and k, = 1, we now divide into ten equal parts, and form the series of powers,

Then a, which lies between b and b+1, must either be equal to one of these powers, or lie between two consecutive powers.

k1

In the former case, x is equal to a fraction k + wherein k1 is one

of the numbers 1, 9.

10'

Wherein k is one of the numbers 0, 1, ... 9. Then z lies between

By continuing the process, we can prove that a either is equal to a

m

It follows, from the nature of the process by which

is obtained,

n

m

that bn increases, but remains always less than a, and therefore becomes

m+1

decreases,

more and more nearly equal to a. For the same reason, b n but is always greater than a, and therefore becomes more and more nearly equal to a.

It will now be proved that as n increases beyond any assigned number,

m

m+1

however great, b" and b n

differ from a by less than any assigned

number, however small.

For, b and bn differ from each other, and hence from a, which lies between them, by less than any assigned number, however small.

m n

m + 1

n

At the same time, as was proved in Ch. XVIII., Art. 7, and differ from each other, and therefore from a common limit which lies between them, by less than any assigned number, however small. This .common limit, therefore, is such a value of x as makes b2 : = a.

We have, therefore, proved that for given values of a and b, when b>1 and a is positive, there is a definite value of x which satisfies the equation

Therefore, there is always a value of x such that b* = a, when b<1 and positive.

Logarithms.

9. The value of x which satisfies the equation b*=a is called the logarithm of a to the base b.

The Logarithm of a given number a to a given base b is, therefore, the exponent of the power to which the base b must be raised to produce the number a.

E.g., since 238, 3 is the logarithm of 8 to the base 2; since 102 = 100, 2 is the logarithm of 100 to the base 10.

10. The relation b2 = 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.

11. The theory of logarithms is based upon the idea of representing all positive numbers, in their natural order, as powers of one and the same base.

Thus, 4, 8, 16, 32, 64, etc., can all be expressed as powers of a common base 2; as 4 = 22, 8 = 23, 16 = 2*, etc. Since, also, all the numbers intermediate between those given above can be expressed as powers of 2, the exponents of these powers are the logarithms of the corresponding numbers.

The logarithms of all positive numbers to a given base form what is called a System of Logarithms. The base is then called the base of the system.

12. Neither the number 1, nor any negative number, can be taken as the base of a system of logarithms. For, since any power of 1 is 1, it is evident that any other number than 1 cannot be represented as a power of 1.

It is also impossible to represent all positive numbers as real powers of a given negative number.

It follows from Art. 8 that any positive number except 1 may be taken as the base of a system of logarithms.

13. The following properties of logarithms evidently follow from the properties of powers.

(i.) The logarithm of 1 to any base is 0. log, 10.

(ii.) The logarithm of the base itself is 1. log, b = 1.

For 6o 1, or

=

For b1=b, or

The following properties of logarithms hold when the base is greater than 1.

(iii.) The logarithm of an infinite is an infinite. For b = ∞, or log, ∞ = ∞.

(iv.) The logarithm of 0 is a negative infinite. For b-"= 0, or log, 0 =

∞.

(v.) The logarithm is positive or negative, according as the number is> or < 1.

For any positive number >1 lies between 1 and +∞. Therefore its logarithm lies between 0 and +∞, and is positive. Any positive number <1 lies between 0 and 1; therefore its logarithm lies between 0, and is negative.

∞ and

14. The following relation will be found useful:

For let log, a = x, or b2 = a.

blogba = a.

Substituting in the last relation log, a for x, we have blog, a = a.

The truth of this relation is self-evident. It asserts that the logarithm of a, to the base b, is the exponent log, a; but the italicized words are just the words for which the expression log, a stands.

EXERCISES I.

Express the following relations in the language of logarithms:

Express the following relations in terms of powers:

9. logs 81 4. 10. log, 812. 11. log4 64=3.

=

13. logs 512 = 3. 14. logs 729=6. 15. log 16-4.

Determine the values of the following logarithms:

12. log264-6.

16. log10.001-3..

20. log2.25.

24. log5.04.

19. log2.5.

23. log2.125.

To the base 16, what numbers have the following logarithms?

15. The logarithm of a product is equal to the sum of the loga rithms of its factors; or,

Let

log。 (m × n) = log。 m + log。 n.

log, m = x and log, n = y;

then bx= m and by : n, and therefore, mn = bxby =bx+y.

Translated into the language of logarithms, this result reads logi (mn) = x + y.