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numbers which have the same significant digits; so that it is sufficient to tabulate the mantissæ of the logarithms of integers.

This proposition we proceed to prove.

411. Let N be any number, then since multiplying or dividing by a power of 10 merely alters the position of the decimal point without changing the sequence of figures, it follows that N X 10P, and N:101, where p and q are any integers, are numbers whose significant digits are the same as those of N. Now log (Nx 10)=log N+p log 10 =log N+P

..(1). Again, log (1:109)=log N-q log 10 =log N-9

..(2). In (1) an integer is added to log N, and in (2) an integer is subtracted from log N; that is, the mantissa or decimal portion of the logarithm remains unaltered.

In this and the three preceding articles the mantissæ have been supposed positive. In order to secure the advantages of Briggs' system, we arrange our work so as always to keep the mantissa positive, so that when the mantissa of any logarithm has been taken from the Tables the characteristic is prefixed with its appropriate sign, according to the rules already given.

412. In the case of a negative logarithm the minus sign is written over the characteristic, and not before it, to indicate that the characteristic alone is negative, and not the whole expression. Thus 4:30103, the logarithm of .0002, is equivalent to - 4+ •30103, and must be distinguished from – 4:30103, an expression in which both the integer and the decimal are negative. In working with negative logarithms an arithmetical artifice will sometimes be necessary in order to make the mantissa positive. For instance, a result such as - 3.69897, in which the whole expression is negative, may be transformed by subtracting 1 from the characteristic and adding 1 to the mantissa. Thus

-3.69897= -4+(1 – 69897)=4:30103. Example 1. Required the logarithm of .0002432.

In Seven-Place Tables we find that 3859636 is the mantissa of log 2432 (the decimal point as well as the characteristic being omitted); and, by Art. 409, the characteristic of the logarithm of the given number is – 4;

log •0002432=2.3859636.

Example 2. Find the value of $-00000165, given

log 165=2.2174839, log 697424=5.8434968. Let x denote the value required; then

1
log x=
x=log (•00000165)5 = log (-00000165)

5

1
( (62174839);

the mantissa of log •00000165 being the same as that of log 165, and the characteristic being prefixed by the rule.

Now $(6_2174839) = {(10 +4.2174839) = 2-8434968

and •8434968 is the mantissa of log 697424; hence x is a' number consisting of these same digits but with one cipher after the decimal point. [Art. 409.] Thus

x=.0697424.

413. To transform logarithms from base a to base b.

Suppose that the logarithms of all numbers to base a are known and tabulated.

Let N be any number whose logarithm to base b is required. Let y=logo N, so that bu=N;

.: loga (W)=loga N; that is,

y logab=loga N;
1

x logan,

y logab

or

1 log, N= x loga N ...............(1).

logab Now since N and b are given, log, N and logab are known from the Tables, and thus logo N may be found. Hence it appears that to transform logarithms from base a to

1 base b we have only to multiply them all by

log. constant quantity and is given by the Tables; it is known as the modulus.

; this is a

x logaa= logab

4

5

100 ~ log 90

Cor. If in equation (1) we put a for N, we obtain

1

1 logra=

logab

.. log,a x logab=1. 414. In Arts. 403-405 we demonstrated,

(1) The logarithm of a product is equal to the sum of the logarithms of its factors.

(2) The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.

(3) The logarithm of any power, integral or fractional, of any quantity is equal to the logarithm of the quantity multiplied by the exponent of the power.

415. The following examples illustrate the application of these principles and the utility of logarithms. Example 1. Given log 3=•4771213, find log {(2.7)3 x (•81)5 -- (90)*}.

27 4 81 5 The required value =3 log

10
+zlog
4

5
=3 (log 33 – 1) + (log 34 – 2)
-5

(log 32 +1)

4
16 5

8 5)
9+
log

+ +
5 2
97

log 3 – 527
10
=4.6280766 – 5.85

= 2.7780766. The student should notice that the logarithm of 5 and its powers can always be obtained from log 2; thus

10
log 5=log=log 10 – log 2=1 – log 2.

2
Example 2. Find the number of digits in 87516, given

log 2=-3010300, log 7=•8450980.
log (87516)= 16 log (7 x 125)

= 16 (log 7+3 log 5)
= 16 (log 7+3 - 3 log 2)
= 16 x 2.9420080

=47072128;
hence the number of digits is 48. [Art. 408.]

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EXAMPLES XL. a.

1. Find the logarithms of 132 and •03125 to base 32, and 100 and .00001 to base .01.

1

2. Find the value of

log,512, log: 0016, logsı 27, log.9343. 3. Write down the numbers whose logarithms to bases

25, 3, 02, 1, –4, 107, 1000
2

-2, -3, 6, - 1, 2, respectively,

1

are

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6. Find by inspection the characteristics of the logarithms of 3174, 625•7, 3:502, 4, 374, .000135, 23.22065.

7. The mantissa of log 37203 is •5705780: write down the logarithms of 37-203, 000037203, 372030000.

8. The logarithm of 7623 is 3.8821259: write down the numbers whose logarithms are •8821259, 6*8821259, 7.8821259.

Given log 2=-3010300, log 3=4771213, log 7=.8450980, find the value of

9. log 729.

10. log 8400.

11. log .256.

+108297

12. log 5.832. 13. log N392. 14. log •3048.

11 490 7 15. Show that log

2 log :=log 2. 15

9 16. Find to six decimal places the value of

225

20

512 log

+log 224

81 17. Simplify log {(10-8)} * (-24)} = (90)-2), and find its numerical value.

-2 log 189

18. Find the value of

log ($126.V108: 21008. V 162). 19. Find the value of log

5

588 x 768
686 X 972

20. Find the number of digits in 4242.

1000

21. Show that

'81
80

is greater than 100000.

22. How many ciphers are there between the decimal point and the first significant digit in ?

1000

23. Find the value of W01008, having given

log 398742=5.6006921. 24. Find the seventh root of •00792, having given log 11=1•0413927 and log 500•977=2.6998179.

75 135 45 25. Find the value of 2 log - +log

49

32

3108 28

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