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From the above it will be seen that, when the base is 10,

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since 10-1 =

since 10-2 =

since 10-3 = 1000 =

.001, the exponent - 3 = log .001; etc. From this it will be seen that the logarithms of exact powers of 10 and of decimals like .1, .01, and .001 are the whole numbers 1, 2, 3, etc. and -1, -2, -3, etc., respectively. Only numbers consisting of 1 and one or more ciphers have whole numbers for logarithms.

Now, it is evident that, to produce a number between 1 and 10, the exponent of 10 must be a fraction; to produce a number between 10 and 100, it must be 1 plus a fraction; to produce a number between 100 and 1,000, it must be 2 plus a fraction; etc. Hence, the logarithm of any number between 1 and 10 is a fraction; of any number between 10 and 100, 1 plus a fraction; of any number between 100 and 1,000, 2 plus a fraction, etc. A logarithm, therefore, usually consists of two parts: a whole number, called the characteristic, and a fraction, called the mantissa. The mantissa is always expressed as a decimal. For example, to produce 20, 10 must have an exponent of approximately 1.30103, or 101. 30103 = 20, very nearly, the degree of exactness depending on the number of decimal places used. Hence, log 20 = 1.30103, 1 being the characteristic, and .30103, the mantissa.

Referring to the second part of the preceding table, it is clear that the logarithms of all numbers less than 1 are negative, the logarithms of those between 1 and .1 being -1 plus a fraction. For, since log .1 = -1, the logarithms of .2, .3, etc. (which are all greater than .1, but less than 1) must be greater than -1; i. e., they must equal -1 plus a fraction. For the same reason, to produce a number between .1 and .01, the logarithm (exponent of 10) would be equal to -2 plus a fraction, and for a number between .01 and .001, it would be equal to -3 plus a fraction. Hence, the logarithm

of any number between 1 and .1 has a negative characteristic of 1 and a positive mantissa; of a number between .1 and .01, a negative characteristic of 2 and a positive mantissa; of a number between .01 and .001, a negative characteristic of 3 and a positive mantissa; of a number between .001 and .0001, a negative characteristic of 4 and a positive mantissa, etc. The negative characteristics are distinguished from the positive by the· sign written over the characteristic. Thus, 3 indicates that 3 is negative.

It must be remembered that in all cases the mantissa is positive. Thus, the logarithm 1.30103 means +1 + .30103, and the logarithm 1.30103 means -1.30103. Were the minus sign written in front of the characteristic, it would indicate that the entire logarithm was negative. Thus, -1.30103 =-1 -.30103.

Rule for Characteristic. -Starting from the unit figure, count the number of places to the first (left-hand) digit of the given number, calling unit's place zero; the number of places thus counted will be the required characteristic. If the first digit lies to the left of the unit figure, the characteristic is positive; if to the right, negative. If the first digit of the number is the unit figure, the characteristic is 0. Thus, the charactertisic of the logarithm of 4,826 is 3, since the first digit, 4, lies in the 3d place to the left of the unit figure, 6. The characteristic of the logarithm of 0.0000072 is-6 or 6, since the first digit, 7, lies in the 6th place to the right of the unit figure. The characteristic of the logarithm of 4.391 is 0, since 4 is both the first digit of the number and also the unit figure.

TO FIND THE LOGARITHM OF A NUMBER.

To aid in obtaining the mantissas of logarithms, tables of logarithms have been calculated, some of which are very elaborate and convenient. In the Table of Logarithms, the mantissas of the logarithms of numbers from 1 to 9,999 are given to five places of decimals. The mantissas of logarithms of larger numbers can be found by interpolation. The table contains the mantissas only: the characteristics may be easily found by the preceding rule.

The table depends on the principle, which will be explained later, that all numbers having the same figures in the same order have the same mantissa, without regard to the position of the decimal point, which affects the characteristic only. To illustrate, if log 206 == 2.31387, then,

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To find the logarithm of a number not having more than four figures:

Rule. Find the first three significant figures of the number whose logarithm is desired, in the left-hand column; find the fourth figure in the column at the top (or bottom) of the page; and in the column under (or above) this figure, and opposite the first three figures previously found, will be the mantissa or decimal part of the logarithm. The characteristic being found as previously described, write it at the left of the mantissa, and the resulting expression will be the logarithm of the required number. EXAMPLE.-Find from the table the logarithm (a) of 476; (b) of 25.47; (c) of 1.073; (d) of .06313.

SOLUTION.-(a) In order to economize space and make the labor of finding the logarithms easier, the first two figures of the mantissa are given only in the column headed 0. The last three figures of the mantissa, opposite 476 in the column headed N (N stands for number), are 761, found in the column headed 0; glancing upwards, we find the first two figures of the mantissa, viz., 67. The characteristic is 2; hence, log 476 = 2.67761.

NOTE.-Since all numbers in the table are decimal fractions, the decimal point is omitted throughout; this is customary in all tables of logarithms.

(b) To find the logarithm of 25.47, we find the first three figures, 254, in the column headed N, and on the same horizontal line, under the column headed 7 (the fourth figure of the given number), will be found the last three figures of the mantissa, viz., 603. The first two figures are evidently 40, and the characteristic is 1; hence, log 25.47 = 1.40603.

(c) For 1.073; in the column headed 3, opposite 107 in the column headed N, the last three figures of the mantissa are found, in the usual manner, to be 060. It will be noticed

that these figures are printed *060, the star meaning that instead of glancing upwards in the column headed 0, and taking 02 for the first two figures, we must glance downwards and take the two figures opposite the number 108, in the left-hand column, i. e., 03. The characteristic being 0, log

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(d) For .06313; the last three figures of the mantissa are found opposite 631, in column headed 3, to be 024. In this case, the first two figures occur in the same row, and are 80. Since the characteristic is 2, log .06313 2.80024.

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If the original number contains but one digit (a cipher is not a digit), annex mentally two ciphers to the right of the digit; if the number contains but two digits (with no ciphers between, as in 4,008), annex mentally one cipher on the right before seeking the mantissa. Thus, if the logarithm of 7 is wanted, seek the mantissa for 700, which is .84510; or, if the logarithm of 48 is wanted, seek the mantissa for 480, which is .68124. Or, find the mantissas of logarithms of numbers between 0 and 100, on the first page of the tables.

The process of finding the logarithm of a number from the table is technically called taking out the logarithm.

To take out the logarithm of a number consisting of more than four figures, it is inexpedient to use more than five figures of the number when using five-place logarithms (the logarithms given in the accompanying table are five-place). Hence, if the number consists of more than five figures and the sixth figure is less than 5, replace all figures after the fifth with ciphers; if the sixth figure is 5 or greater, increase the fifth figure by 1 and replace the remaining figures with ciphers. Thus, if the number is 31,415,926, find the logarithm of 31,416,000; if 31,415,426, find the logarithm of 31,415,000.

EXAMPLE.-Find log 31,416.

SOLUTION.-Find the mantissa of the logarithm of the first four figures, as explained above. This is, in the present case, .49707. Now, subtract the number in the column headed 1, opposite 314 (the first three figures of the given number), from the next greater consecutive number, in this case 721, in the column headed 2. 721-707 14: this number is called the difference. At the extreme right of the page will be found a

secondary table headed P. P., and at the top of one of these columns, in this table, in bold-face type, will be found the difference. It will be noticed that each column is divided into two parts by a vertical line, and that the figures on the left of this line run in sequence from 1 to 9. Considering the difference column headed 14, we see opposite the number 6 (6 is the last or fifth figure of the number whose logarithm we are taking out) the number 8.4, and we add this number to the mantissa found above, disregarding the decimal point in the mantissa, obtaining 49,707 + 8.4 = 49,715.4. Now, since 4 is less than 5, we reject it, and obtain for our complete mantissa .49715. Since the characteristic of the logarithm of 31,416 is 4, log 31,416 = 4.49715.

EXAMPLE.-Find log 380.93.

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SOLUTION.-Proceeding in exactly the same manner as above, the mantissa for 3,809 is 58,081 (the star directs us to take 58 instead of 57 for the first two figures); the next greater mantissa is 58,092, found in the column headed 0, opposite 381 in column headed N. The difference is 092-081 11. Looking in the section headed P. P. for column headed 11, we find opposite 3, 3.3; neglecting the .3, since it is less than 5, 3 is the amount to be added to the mantissa of the logarithm of 3,809 to form the logarithm of 38,093. Hence, 58,081 + 3 = 58,084, and since the characteristic is 2, log 380.93 = 2.58084.

EXAMPLE.-Find log 1,296,728.

SOLUTION.-Since this number consists of more than five figures and the sixth figure is less than 5, we find the loga rithm of 1,296,700 and call it the logarithm of 1,296,728. The mantissa of log 1,296 is found to be 11,261. The difference is 294-261 = 33. Looking in the P. P. section for column headed 33, we find opposite 7, on the extreme left, 23.1; neg. lecting the .1, the amount to be added to the above mantissa is 23. Hence, the mantissa of log 1,296,728 11,261 +23 11,284; since the characteristic is 6, log 1,296,728 = 6.11284.

EXAMPLE.-Find log 89.126.

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SOLUTION.-Log 89.12 1.94998. Difference between this and log 89.13 1.95002-1.94998 = 4. The P. P. (propor tional part) for the fifth figure of the number 6 is 2.4, or 2. Hence, log 89.126 1.94998.00002 1.95000.

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