3. Logarithms with base 9 to logarithms with base 3. Ans. Multiply each logarithm by 2.

4. Find log,8, log, 1, log, 2, log,1, log‰ 128.

66. Tables of Logarithms. The common logarithms of all integers from 1 to 100000 have been found and registered in tables, which are therefore called tabular logarithms. In most tables they are given to six places of decimals, though they may be calculated to various degrees of approximation, such as five, six, seven, or a higher number of decimal places. Tables of logarithms to seven places of decimals are in common use for astronomical and mathematical calculations. The common system to base 10 is the one in practical use, and it has two great advantages:

(1) From the rule (Art. 64) the characteristics can be written down at once, so that only the mantissæ have to be given in the tables.

(2) The mantissæ are the same for the logarithms of all numbers which have the same significant digits, in the same order, so that it is sufficient to tabulate the mantissæ of the logarithms of integers.

For, since altering the position of the decimal point without changing the sequence of figures merely multiplies or divides the number by an integral power of 10, it follows that its logarithm will be increased or diminished by an integer; i.e., that the mantissa of the logarithm remains unaltered.

In General. If N be any number, and p and q any integers, it follows that N x 10o and N ÷ 10 are numbers whose significant digits are the same as those of N.

Then log (Nx 10") = log N + plog 10 = log N+p. (1)

Also, log (N÷10')= log Ng log 10= log N-q. (2)

In (1) the logarithm of N is increased by an integer, and in (2) it is diminished by an integer.

That is, the same mantissa serves for the logarithms of all numbers, whether greater or less than unity, which have the same significant digits, and differ only in the position of the decimal point.

This will perhaps be better understood if we take a particular case.

From a table of logarithms we find the mantissa of the logarithm of 787 to be 895975; therefore, prefixing the characteristic with its appropriate sign according to the rule, we have

Also,

log 78700 log (787 x 100)= log 787 +2

=

= 4.895975.

NOTE 1. We do not write log10 787; for so long as we are treating of logarithms to the particular base 10, we may omit the suffix.

NOTE 2.- - Sometimes in working with negative logarithms, an arithmetic artifice will be necessary to make the mantissa positive. For example, a result such as -2.69897, in which the whole expression is negative, may be transformed by subtracting 1 from the characteristic, and adding 1 to the mantissa. Thus,

-2.698973 + (1 - .69897) = 3.30103.

NOTE 3. When the characteristic of a logarithm is negative, it is often, especially in Astronomy and Geodesy, for convenience, made positive by the addition of 10, which can lead to no error, if we are careful to subtract 10.

Thus, instead of the logarithm 3.608582, we may write 7.603582 - 10.

In calculations with negative characteristics we follow the rules of Algebra.

the 1 carried from the last subtraction in decimal places changes 5 into - 4, and then

gives 1 as a result.

3. Multiply 2.1528 by 7.

4 subtracted from

3

2.1528
7

13.0696

the 1 carried from the last multiplication of the decimal places being added to - 14, and thus giving 13 as a result.

NOTE 4. - When a logarithm with negative characteristic has to be divided by a number which is not an exact divisor of the characteristic, we proceed as follows in order to keep the characteristic integral. Increase the characteristic numerically by a number which will make it exactly divisible, and prefix an equal positive number to the mantissa.

4. Divide 3.7268 by 5.

Increase the negative characteristic so that it may be exactly divisible by 5; thus

Given that log2=.30103, log 3=.47712, and log 7=.84510; find the values of

5. log 6, log 42, log 16. Ans. .77815, 1.62325, 1.20412.

6. log 49, log 36, log 63. Ans. 1.69020, 1.55630, 1.79934.

7. log 200, log 600, log 70.

2.30103, 2.77815, 1.84510.

8. log 60, log.03, log 1.05, log.0000432.

NOTE. The logarithm of 5 and its powers can always be obtained from log 2.

Ans. 1.77815, 2.47712, .02119, 5.63548.

9. Given log2=.30103; find log 128, log 125, and log 2500. Ans. 2.10721, 2.09691, 3.39794.

Given the logarithms of 2, 3, and 7, as above; find the logarithms of the following:

10. 20736, 432, 98, 686, 1.728, .336.

Ans. 4.31672, 2.63548, 1.99122, 2.83632, .23754, 1.52634.

11. √.2, (.03)1, (.0021)*, (.098)3, (.00042)3, (.0336)1. Ans. 1.65052, 1.61928, 1.46444, 4.97368, 17.11625, 1.26317.

67. Use of Tables of Logarithms * of Numbers. — In our explanations of the use of tables of common logarithms we shall use tables of seven places of decimals.† These tables are arranged so as to give the mantissæ of the logarithms of the natural members from 1 to 100000; i.e., of numbers containing from one to five digits.

A table of logarithms of numbers correct to seven decimal places is exact for all the practical purposes of Astronomy and Geodesy. For an actual measurement of any kind must be made with the greatest care, with the most accurate instruments, by the most skilful observers, if it is to attain. to anything like the accuracy represented by seven significant figures.

* The methods by which these tables are formed will be given in Chap. VIII. The student should here provide himself with logarithmic and trigonometric tables of seven decimal places. The most convenient seven-figure tables used in this country are Stanley's, Vega's, Bruhns', etc. In the appendix to the Elementary Trigonometry are given five-figured tables, which are sufficiently near for most practical applications.

If the measure of any length is known accurately to seven figures, it is practically exact; i.e., it is known to within the limits of observation.

If the measure of any angle is known to within the tenth part of a second, the greatest accuracy possible, at present, in the measurement of angles is reached. The tenth part of a second is about the twomillionth part of a radian. This degree of accuracy is attainable only with the largest and best instruments, and under the most favorable conditions.

On page 101 is a specimen page of Logarithmic Tables. It consists of the mantissæ of the logarithms, correct to seven places of decimals, of all numbers between 62500 and 63009. The figures of the number are those in the left column headed N, followed by one in larger type at the top of the page. The first three figures of the mantissæ 795, 796, 797 ..., and the remaining four are in the same horizontal line with the first four figures of the number, and in the vertical column under the last.

Logarithms are in general incommensurable numbers. Their values can therefore only be given approximately. Throughout all approximate calculations it is usual to take for the last figure which we retain, that figure which gives the nearest approach to the true value. When only a certain number of decimal places is required, the general rule. is this: Strike out the rest of the figures, and increase the last figure retained by 1 if the first figure struck off is 5 or greater than 5.

68. To find the Logarithm of a Given Number. When the given number has not more than five digits, we have merely to take the mantissa immediately from the table, and prefix the characteristic by the rule (Art. 64).

Thus, suppose we require the logarithm of 62541. The table gives .7961648 as the mantissa, and the characteristic is 4, by the rule; therefore

Similarly,

log 62541

= 4.7961648.

log.0062813.7980288

(Art. 64)