CHAPTER XX

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LOGARITHMS

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207. Generalized powers. If b and c are integers, we can easily compute b. When c 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 V8, 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

51, 514, 5141, 51.414,....

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The computation of such a number as 5141 would be somewhat laborious, but could be performed, since 5141 — 5188514. Thus it is a root of the equation 2100 Horner's method, p. 197.

5141, and could be found by

We see in this particular case that 5 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 b is a positive integer and c 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 bea. We now consider the case where a and b are given and c remains to be found. Let

= 8,b=2. Then if 2o = 8, we see immediately that c = 3 satisfies this equation. If a = 16, b 2, then 2o = 16 and c = 4 is the solution. If a 2, consider the equation 2o

10, = b

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: 10. If we

let c = = 3, we see that 23 8. If we let c equal the next larger integer, 4, we see 24 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, 1 128,

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

(a) log 49 = 2.

(c) log 225 = 2.

(b) log 81 = 4.

(d) log 625 = 4.

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.

THEOREM III. The logarithm of the quotient of two numbers is the difference between the logarithms of the numbers.

THEOREM IV. The logarithm of the real nth root of a number is the logarithm of the number divided by n.

By Theorem I, log (73. √5)= log √73+ log √5.

* Where no base is written it is assumed that the base 10 is employed.

210. Common system of logarithms. For purposes of computation 10 is taken as a base, and unless some other base is indicated we shall assume that such is the case for the rest of this chapter. We may write as follows the equations which show the numbers of which integers are the logarithms.

Assuming that as x becomes greater log x also becomes greater, we see that a number, for example, between 10 and 100 has a logarithm between 1 and 2. In fact the logarithm of any number not an exact power of 10 consists of a whole-number part and a decimal part.

The whole-number part of the logarithm of a number is called the characteristic of the logarithm.

The decimal part of the logarithm of a number is called the mantissa of the logarithm.

The characteristic of the logarithm of any number may be seen from the above table, from which the following rules are immediately deduced.

The characteristic of the logarithm of a number greater than unity is one less than the number of digits to the left of its decimal point.

Thus the characteristic of the logarithm of 471 is 2, since 471 is between 100 and 1000; of 27.93 is 1, since this number is between 10 and 100; of 8964.2 is 3, since this number is between 1000 and 10,000.