Imágenes de páginas
PDF
EPUB
[blocks in formation]

the exponent L is called the logarithm of the number N, to the

base 10.

For example,

102 =

100, hence 2 is the logarithm of 100, to the base 10.

We write, 2 = log 100.

108 1000, hence 3 is the logarithm of 1000, to the base 10.

=

We write, 3 = log 1000.

10 = 3.16227+ (verify this), hence .5 is the logarithm of 3.16227+, to the base 10. We write, .5= log 3.16227+.

10 = 101(10)

=

the base 10.

- 31.6227+, hence 1.5 is the logarithm of 31.6227+, to We write, 1.5 = log 31.6227+.

As in this chapter all logarithms are taken to the same base 10, no confusion will arise from the omission, hereafter, of the phrase "to the base 10."

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

108. On a piece of square paper as small as that in Fig. 15, it is not convenient to draw a curve that will exhibit the logarithms of numbers ranging from .001, the smallest, to 1000, the largest number considered above. The curve shows the logarithms of positive numbers between .1 and 50. The numbers are laid off on the x-axis, their respective logarithms on the y-axis.

[blocks in formation]

1.5

log 31.6+ locates the point E

In drawing the curve, some additional values were used.

X 50

ORAL EXERCISES ON THE LOGARITHMIC CURVE

109. 1. Where does the curve cut the x-axis? How much is log 1?

2. What is the algebraic sign of the logarithms of all numbers larger than 1?

3. For what range of numbers are the logarithms negative? 4. Does the logarithmic curve extend to the left of the y-axis? 5. Do negative numbers have real logarithms ?

6. Does the logarithm increase as a variable number increases?

By inspection of Fig. 15, find approximately the logarithms of the following numbers:

[blocks in formation]

By inspection of Fig. 15, find approximately the numbers corresponding to the following logarithms:

[blocks in formation]
[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

110. In drawing the logarithmic curve in Fig. 15 we located a few points and then drew a smooth curve through those points. This process is based on the following assumptions which admit of proof: (1) All positive numbers, whether rational or irrational, have logarithms; (2) In all cases, the logarithm of a number increases when the number itself increases.

FUNDAMENTAL THEOREM

111. In studying the logarithmic curve we noticed that log 50 log 5 + log 10.

=

This illustrates an important theorem.

Let N and N1 be any two positive numbers.

[blocks in formation]

The logarithm of the product of two positive numbers is equal to the sum of the logarithms of the numbers.

112. The integral part of a logarithm is called its characteristic, and the decimal part is called its mantissa.

For example, log 31.6227+ = 1.5.

The characteristic of log 31.6227+ is 1.
The mantissa of log 31.6227+ is .5.

The tables of logarithms give only the mantissa; the characteristic can be supplied by two easy rules.

It has been found convenient to take the mantissas of all logarithms positive. The characteristic is positive in the logarithms of numbers larger than 1, and negative in the logarithms of numbers smaller than 1. That is, the characteristic of log 100 is positive; the characteristic of log .01 is negative. If the characteristic is negative, the negative sign is placed over the figure, as a reminder that the does not apply to the mantissa. Thus the characteristic of log .06 is 2.

For the purpose of deriving the rule for determining the characteristic of the logarithm of a number, we restate some of the relations in § 107:

[blocks in formation]

Since 569.5 lies between 100 and 1000, its logarithm lies between 2 and 3.

That is,

Since 85.6 lies between 10 and 100,
Since 7.03 lies between 1 and 10,
Since .673 lies between .1 and 1,
Since .045 lies between .01 and .1,

log 569.5 = 2 + a mantissa.

log 85.61 + a mantissa.
log 7.030+ a mantissa.
log .673-1+ a mantissa.
log .045 =
== - 2+ a mantissa.
3+ a mantissa.

Since .0078 lies between .001 and .01, log .0078

By inspection of these relations we obtain the rule:

1. If the first significant figure of a number is n places to the

right of units' place, the characteristic of the logarithm is

+

n

« AnteriorContinuar »