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CHAPTER V.

LOGARITHMS.

20. DEF.—The logarithm of a number to a given base is the index of the power to which the base must be raised to obtain the number.

Thus, we may obtain the numbers 1, 10, 100, 1,000, 10,000, &c., by raising the base 10 to the powers 0, 1, 2, 3, 4, &c., respectively; and hence, by the above definition, we have

Log 1 = 0, log 10 1, log 100 = 2, log 1,000 3, &c.,

10

10

10

10

the suffix 10 being added to the word log to indicate that the base is 10. It is usual, however, in common logarithms to omit this suffix; and hence, when there is no base expressed, the student will understand 10.

Again, the numbers 1, 2, 4, 8, 16, &c., may be obtained by finding the values of 20, 21, 22, 23, 24, &c., respectively, and hence we have by definition

0, log 2 = 1, log 4'= 2, log 8 = ,

Log 1

2

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4

5

3

36

So also we find log 16 = 2, log 125 =

3, log 81 = 4,&c. Ex. Find log 256, log 216, and the logarithm of 9 to base 13.

Log 256 = log 4* = 4, by definition.
Log 216 = log 68 = log (69* = log 36* = $, by definition .
Log 9 = log 3* = log (13)' = 4, by definition.

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Characteristics of Ordinary Logarithms. 21. DEF.—The characteristic of a logarithm is the integral part of the logarithm, and the fractional part (generally expressed as a decimal) is called the mantissa.

10n + 12

1. Numbers containing integer digits.

Every number containing n digits in its integral part must lie between 10n-1 and 10”.

Thus, 6 lies between 10° and 101, 29 lies between 101 and 10%, 839 lies between 10% and 10%, &c.

Hence the ordinary logarithms of all numbers having n integer digits lies between (n − 1) and n.

The integral portion or characteristic of the logarithm of a number having n integer digits is therefore (n 1).

Hence we have the following rule :

RULE 1.—The characteristic of the logarithm of a number having integer digits is one less than the number of integer digits.

Thus, the characteristics of the logarithms of 32, 713.54, 8.7168, 56452, 73607.9 are respectively 1, 2, 0, 4, 4.

2. Numbers less than unity expressed as decimals. All such numbers having n zeros immediately after the

1

1 decimal point lie between and

or between 10-9

10n and 10-(n+1).

1 Thus, •3 lies between 1 and 'l, or between 1 and or 10° and

' 10- 1;

1 ·027 lies between •1 and .01, or between and

10 and 10- ';

1

1 000354 lies between .001 and .0001, or between and

103 or 10 – 3 and 10-4, and so on.

Hence, by Def., Art. 20, the logarithm of any number having n zeros immediately after the decimal point lies between

n and - (n + 1). Hence, the logarithm is negative, and the integral part of this negative quantity is n. It is however usual to write all the mantissæ of logarithms as positive quantities, and the negative integral part of the logarithm will be the next higher negative integer, viz., – (n + 1).

We have therefore the following rule :

RULE 2.-The.characteristic of the logarithm of a number less than unity, and expressed as a decimal, is the negativa

or 10-1

102

104'

integer next greater than the number of zeros immediately after the decimal point.

Thus, the characteristics of the logarithms of 3, .0076, 02535, 7687, are respectively - 1, -3, - 2, – 1.

22. The logarithm of the PRODUCT of two numbers is the SUM of the logarithms of the numbers.

Let m and n be the numbers, and let a be the base. Since m and n must be each some power of a, integral or fractional, positive or negative, assume

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Now we have mn = a.ar at+y, and hence, by definition,

log, mn = x + y; we therefore haveloga (mn)

log, m + loga n. Q.E.D. Cor. This proposition may be extended to any number of factors.

Thus, loga (mnpg) = loga m + loga n + loga p + loga 9.

23. The logarithm of the QUOTIENT of two numbers is found by SUBTRACTING the logarithm of the denominator from the logarithm of the numerator. Assuming, as in the last Art., we have

log, m, y

log, n.
a*
Also,

-, and hence, by definition,

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m

at

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m

a n

log,

log, m log, n. Q.E.D. 24. The logarithm of the POWER of a number is found by MULTIPLYING the logarithm of the number by the INDEX of the power.

log, n.

log, NP

Let it be required to find log, No.
Assume N = a“, and therefore a =
We have NP (at)s aps, and hence, by definition,

1$ = px = p log, N. Q.E.D. Ex. 1.-Given log 2 = 3010300, and log 3 = .4771213, find the logarithms of 18, 15, 125, 675.

Log 18 = log (2 x 3) = log 2 + 2 log 3 = 3010300 + 2(4771213) 1.2552726.

Log 15 = log (3 x V) = log 3 + log 10 - log 2 = -4771213 + 1 - 3010300 1.1760913.

Log •125 = log(a) = log1

(2:4)*

= 0 - 3 x 3010300 23 .9030900 1 + (1 - •9030900) 1 + :0969100, or, as usually written, = 1.0969100.

27 33 Log 6.75 = log

4

log na = 3log 3 – 2 log 2 = 3(14771213) – 27.3010300) = .8293039. Ex. 2. Find the logarithm of

(375)

having

(2-43)* * (-032) given log 2 and log 3. We have

1 log 2-4 + 4 log 375 - 5 log 2.43

- (- }) log .032.
1 23 x 3

25
+ 4 log – 5 log ina + log
10

103
(3 log 2 + log 3 - log 10) + 4 (log 3 - 3 log 2)
- 5 (5 log 3 – 2 log 10) + } (5 log 2 - 3 log 10)
(1 - 12 + f) log 2 + (1 + 4 - 25) log 3

+ (- 1 + 10 - 1) log 10 - 573 3010300

4771213 + 4 x 1 2.6590983 9.7809867 + 8.5 = 3.9400850 = 2 + (4 - 3.9400850) = 4 0599150.

Log N

log

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Ex. IV. 1. Find the logarithm to base 4 of the following numbers: 16, 64, 2, 25, .0625, 8. 2. Find the value of log 32, log 25, log 729.

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3. Given log 2 = 3010300, and log 3 = .4771213, find the logarithms of 12, 36, 45, 75, •04, 3.75, -6, ••74.

4. Given log 20763 = 4.3172901, what is the logarithm of 2.0763, 2076-3, .020763, 0020763?

5. Write down the characteristics of the common logarithms of 29-6, -25402, .0034, 6176.003.

6. Given log 20.912 1.3203956, what numbers correspond to the following logarithms :-2.3203956, 6-3203956, 1.3203956, 4:3203956 ?

7. Given log 20.713 = 1:3162430, and log 20714 = 3-3162640, find log .2071457.

8. Given log 3.4937 •5432856, and log 3.4938 = •5432980, find the number whose logarithm is 3.5432930. 9. Given log 1.05 = .0211893, log 2.7 =

·0211893, log 2.7 = 1.4313638, log

2-1303338, find the value of log (2-7)* * 913-5

135 =

(1:05) 10. Given log 18 = 1.2552725, and log 2.4

24 = •3802112, find the value of log •00135. 11. What are the characteristics of log 1167, and log 1965? 12. Having log 2 = •3010300, and log 3 = .4771213, find 2 when 18+ 125.

CHAPTER VI.

THE USE OF TABLES.

25. Tables have been formed of the logarithms of all numbers from 1 to 100,000, and we shall now show how they are

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