3 is the characteristic therefore, and 4976206 the mantissa of log 3145.

116. The following remarks may render more evident the reasoning of the important proposition which comes after them.

The least integer consisting of n digits is 10"-1; for instance, the least integer of four digits is 1000, which is 103. Any other integer consisting of n digits, with or without a decimal fraction attached, is between 10"-1 and 10"; thus 67642, which contains five digits, is between 10,000 and 100,000, that is, between 104 and 105; and 123456-421 is between 100,000 and 1,000,000, or between 105 and 106.

1 1 10,000 104

=

Again, the least decimal fraction which has n ciphers immediately after the decimal point is 10-(+1); for instance, the least decimal with three ciphers thus placed is .0001 = = 10-4. Any other decimal fraction, with n ciphers immediately following the point, is between 10-(+1) and 10-"; thus 0005 is between 0001 and '001, that is, between 10-4 and 10-3; and so in all similar cases.

The student, if acquainted with the theory of Notation in Algebra, may prove generally the above statements; if not, he may without difficulty satisfy himself of their truth by simple reflection.

117. To determine the characteristic of the logarithm of any number.

This can always be done by inspection of the number itself, as will be shown.

(i.) Let the number be greater than unity, and consist of an integer of n digits with or without a decimal fraction. It is then either 10-1, or between 10^-1 and 10". Its logarithm is therefore either n― 1 or between n-1 and n. Therefore the characteristic of its logarithm is n-1. Hence the rule,—

The characteristic of the logarithm of a number greater

than unity is less by one than the number of digits in the integral part of the number.

(ii.) Let the number be a decimal fraction. Since log 10, the logarithms of all decimal fractions will be negative. It is found convenient in practice to have the characteristic only of these logarithms negative, and the mantissa positive. Remembering this, we shape our reasoning as follows:

Let there be n ciphers immediately after the point. It is then either 10-(+1), or between 10-(+1) and 10−”. Its logarithm is therefore either (n+1), or, preserving the mantissa positive, −(n+1)+a fraction. Therefore the characteristic of its logarithm is (n+1). Hence the rule,—

The characteristic of the logarithm of a decimal fraction is greater by one than the number of ciphers which immediately follow the point, and is negative.

118. It would perhaps assist the memory to incorporate the two rules given above into one, thus :

If the number be greater than unity, take the number of its integral digits positively; if it be a decimal fraction, take the number of ciphers which immediately follow the point negatively; subtract one in each case for the characteristic of its logarithm.

119. In the logarithms of decimal fractions the minus sign is placed above the characteristic to denote that it applies only to the characteristic, and not to the mantissa, which, as before said, is always preserved positive. 120. Significant and non-significant digits.

Take the number

56.0802,

:

and either multiply or divide it by any power of 10:for instance, multiply it by 103, 104, and by 106; and divide it by 102 and 105; and we obtain

(1) 56080-2, (2) 560802, (3) 56080200, (4) ·560802, (5) 000560802;

placing the decimal point in (2) and (3), where ordinarily it would be merely understood.

In all the above numbers, the digits 5, 6, 0, 8, 0, 2 appear in order, while in two of them, (3) and (5), some additional ciphers occur. These latter are merely used to fix the position of the decimal point, when by the multiplication or division it is moved away from the former digits to the right or left.

The digits which always occur in such sets of numbers as the above are called significant digits; whilst the additional ciphers which are used occasionally, and merely to find the position of the decimal point, are called non-significant digits.

It will appear from instances such as the above that, N being any number,

N, Nx10", and N÷10",

where n is a positive integer, all contain the same significant digits in the same order.

121. All numbers which have the same significant figures in the same order, have also the same mantissa in their logarithms; that is,

log N, log N. 10", and log

N

10"

all have the same mantissa when n is a positive integer.

and the decimal part of log N cannot be altered either by adding the whole number n as in (1), or subtracting it as in (2). Therefore

N log N, log N. 10", and log

10"

have the same mantissa.

122. Examples will now follow on the foregoing.

(1.) Find the logarithm of 16 5/16 to the base 3/2.

Let x =

· required logarithm. Then, by definition of a logarithm,

transposing, x (2 log 3-log 2) = 1 + 2 log 3

In order to find log 18, log 2 will be required; and log 2 may be obtained from log 25, which is known from log 2.5.

In these examples the two following logarithms are frequently "given." To save repetition, their values are placed here:

log 2 = 30103, log 3

1. Find, by inspection of the following numbers, the characteristics of their logarithms :-3146, 3146, 200, ·002, 3·05, ·0000101, 101-01, 1876-4, 1·02.

2. Which are the significant digits in 3060, 05642, 841, 0001, 11000, and 11001.

3. Prove log (35 × 43)

=

5 log 3+ 6 log 2.

4. Why have log10 3.56 and log10 0356 the same mantissa, and log10 0356 and logio 04 the same characteristic ?

5. Write out the logarithms of ∙117, 117, and ⚫000117.

Given log 1.17 = .06818.

6. Find log, 256 and log 256 to base 2/2.

7. What are the logarithms of 100 and 001 to base 10, and of 1000 to base '01 ?

8. What is the logarithm of 813/3 to the base 3/3?

9. Find the logarithm of a to the base a".

10. Show that log, N

3 logs N.

11. Of what number is -5 the logarithm to base 10?

12. Find, without tables and to three places of decimals, the numbers of which 1.5, 3, and 1-3 are the logarithms in common logarithms.

13. What is the base of the system in which log 10=2?

14. Prove that (log z)log* = 1, when x=1.

15. If the logarithms of numbers in the ordinary tables were all doubled, what would be the base to which they would then be the logarithms of the same numbers as before?