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011-3-29 H.C.M



1. Powers of 10. Consider the following table of values of powers of 10:

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This table may be used for multiplying or dividing powers of 10, by means of the rules 10a. 10b = 10a+b, 10a 106 = 10a-b. Thus, to multiply 1000 by 100,000, add the exponent of 10 in column A opposite 1000 to the exponent of 10 opposite 100,000: 3+58; and look for the number in column B opposite 108, i.e. 100,000,000. Similarly 1,000,000 × .0001 since 6+ (-4) = 2.


To divide 1,000,000 by 100, from the exponent of 10 opposite 1,000,000 subtract the exponent of 10 opposite 100; 6 − 2 = 4; and look for the number opposite 104, i.e. 10,000. Similarly .001÷1,000,000.000000001, since - 3-6-9. To find the 4th power of 100, multiply the exponent of 10 opposite 100 by 4: 4 x 2 = 8, and look for the number opposite 108, i.e. 100,000,000. Likewise (.001)3 = .000000001, since 3 × (− 3) 9. To find the cube root of 1,000,000,000, divide the exponent of 10 opposite 1,000,000,000 by 3, 933, and look for the number opposite 103.


* This Explanation, written to accompany the five-place tables, may be used also for the four-place tables by omitting the last figure in each example in a manner obvious to the teacher.


2. Common Logarithms. The exponent of 10 in any row of column A is called the common logarithm* of the number opposite in column B; thus log 10 = 1, log 100 = 2, log 1000 = 3, etc.; log 1 = 0, log .1=−1; log .01 == 2, log .001 =- 3, etc. In general, if 10' =n, l is called the common logarithm of n, and is denoted by log n.

3. Fundamental Principles. Logarithms are useful in reducing the labor of performing a series of operations of multiplication, division, raising to powers, extracting roots, as above; they have no necessary connection with trigonometry, since all the operations could be performed without them; but they are a great labor-saving device in arithmetical computations. They do not apply to addition and subtraction.

The principles of their application are stated as follows:


I. The logarithm of a product is equal to the sum of the logarithms of the factors: log ab = log a + log b. This follows from the fact that if =a and 104 = b, 101+L: = a·b. In brief: to multiply, add logarithms. II. The logarithm of a fraction is equal to the difference obtained by subtracting the logarithm of the denominator from the logarithm of the numerator: log (a/b) = log a — - log b. For, if 10: =a and 10 = b, then 10-L- ab. In brief: to divide, subtract logarithms.

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III. The logarithm of a power is equal to the logarithm of the base multiplied by the exponent of the power: log ab This follows from the the fact that if 10' = a, then 10 = ab.

IV. The logarithm of a root of a number is found by dividing the logarithm of the number by the index of the root: log Va = (log a)/b. This follows from the fact that if 10 = a, then 10% = a1 – Va.

Corollary of II. The logarithm of the reciprocal of a number is the negative of the logarithm of the number: log (1/a): == - log a, since log 1 = 0.

4. Characteristic and Mantissa. It is shown in algebra that every real positive number has a real common logarithm, and that if a and b are any two real positive numbers such that ab, then log a log b. Neither zero nor any negative number has a real logarithm.

An inspection of the following table, which is a restatement of a part

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*Common logarithms are exponents of the base 10; other systems of logarithms have bases different from 10; Napierian logarithms (see Table VII, p. 112) have a base denoted by e, an irrational number whose value is approximately 2.71828. When it is necessary to call attention to the base, the expression log10 n will mean common logarithm of n; loge n will mean the Napierian logarithm, etc.; but in this book log n denotes log1on unless otherwise

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the logarithm of every number between 1 and 10 is a proper fraction, the logarithm of every number between 10 and 100 is 1 + a fraction, the logarithm of every number between 100 and 1000 is 2 + a fraction; and so on. It is evident that the logarithm of every number (not an exact power of 10) consists of a whole number + a fraction (usually written as a decimal). The whole number is called the characteristic ; the decimal is called the mantissa. The characteristic of the logarithm of any number greater than 1 may be determined as follows:

RULE I. The characteristic of any number greater than 1 is one less than the number of digits before the decimal point.

The following table, which is taken from § 1, p. v, shows that


log a


.00001 .0001 .001
- 5




- 3



- 1


.0000001 - 7 - 6 the logarithm of every number between .1 and 1 is the logarithm of every number between .01 and .1 is the logarithm of every number between .001 and .01 is — 3+ a fraction; and so on.

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1 + a fraction,

2+ a fraction,

Thus the characteristic of every number between 0 and 1 is a negative whole number; there is a great practical advantage, however, in computing, to write these characteristics as follows: 19-10, − 2 = 8 — 10, - 37 10, etc. E.g. the logarithm of .562 is - 1.74974, but this should be written 9.74974 - 10; and similarly for all numbers less than 1. RULE II. The characteristic of a number less than 1 is found by subtracting from 9 the number of ciphers between the decimal point and the first significant digit, and writing 10 after the result.

Thus, the characteristic of log 845 is 2 by Rule I; the characteristic of log 84.5 is 1 by (I); of log 8.45 is 0 by (I); of log.845 is 9 - 10 by (II); of log.0845 is 8 - 10 by (II).

An important consequence of what precedes is the following:

To move the decimal point in a given number one place to the right is equivalent to adding one unit to its logarithm, because this is equivalent to multiplying the given number by 10. Likewise, to move the decimal point one place to the left is equivalent to subtracting one unit from the logarithm. Hence, moving the decimal point any number of places to the right or left does not change the mantissa but only the characteristic.* Thus, 5345, 5.345, 534.5, .05345, 534500 all have the same mantissa.

*Another rule for finding the characteristic, based on this property, is often useful: if the decimal point were just after the first significant figure, the characteristic would be zero; start at this point and count the digits passed over to the left or right to the actual decimal point; the number obtained is the characteristic, except for sign; the sign is negative if the movement was to the left, positive if the movement was to the right.

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