multiple roots, we may use all the previous methods simultane

ously. Hence for this case we assume the expansion

Adding (2) and (5), (1) and (4), we have, together with (3),

Substituting in (3) and (7) and solving, we find C = 3, B = 2. Substituting in (1), we find D : =2. Similarly, from (5), E = 6.

CHAPTER XX

LOGARITHMS

207. Generalized powers. If b and c are integers, we can easily compute b. When e is not an integer but a fraction we can compute the value of be to any desired degree of accuracy. Thus if b = 2, c = 3, we have 2 = √28 = √8, which we can find to any number of decimal places. If, however, the exponent is an irrational number as √2, we have shown no method of computing the expression. Since, however, V2 was seen (p. 55) to be the limit approached by the sequence of numbers

1, 1.4, 1.41, 1.414,,

it turns out that 5 is the limit approached by the numbers

5, 54, 5141, 51414,

=

100

The computation of such a number as 5141 would be somewhat laborious, but could be performed, since 5141515141. Thus it is a root of the equation x100 Horner's method, p. 197.

=

5141, and could be found by

We see in this particular case that 52 is the limit approached by a sequence of numbers where the exponents are the successive approximations to √2 obtained by the process of extracting the square root. In a similar manner we could express the meaning of be, where is a positive integer and e is any irrational number.

ASSUMPTION. We assume that the laws of operation which we have adopted for rational exponents hold when the exponents are irrational.

where c and d are any numbers, rational or irrational.

208. Logarithms. We have just seen that when b and c are given a number a exists such that be = a. We now consider the case where a and b are given and c remains to be found. Let α = 8,b=2. Then if 2o 8, we see immediately that c = 3 satis2, then 2o 16 and c = =

fies this equation. If a = solution. If a = 10, b

let c =

=

=

3, we see that 23

integer, 4, we see 2

=

=

=

16, b
2, consider the equation 2o

=

4 is the

10. If we

8. If we let c equal the next larger

16. If then any number c exists such that

2o 10, it must evidently lie between 3 and 4. To prove the existence of such a number is beyond the scope of this chapter, but we make the following

ASSUMPTION. There always exists a real number x which satisfies the equation

where a and b are positive numbers, provided b 1.

(1)

Since any real number is expressible approximately in terms of a decimal fraction, this number x is so expressible.

The power to which a given number called the base must be raised to equal a second number is called the logarithm of the second number.

In (1) x is the logarithm of a for the base b.

This is abbreviated into

The number a in (1) and (2) is called the antilogarithm.

(2)

EXERCISES

1. In the following name the base, the logarithm, and the antilogarithm, and write in form (2).

2. Find the logarithms of the following numbers for the base 3: 81, 243, 1, 1, 81.

3. For base 2 find logarithms of 8, 128, 1, 1

4. What must the base be when the following equations are true?

209. Operations on logarithms. By means of the law expressed in the Assumption, § 207, we arrive at principles that have made the use of logarithms the most helpful aid in computations that is known.

THEOREM I. The logarithm of the product of two numbers is the sum of their logarithms.

THEOREM II. The logarithm of the nth power of a number is n times the logarithm of the number.