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Logarithms are used to simplify numerical computations involving multiplications, di- vision, powers and roots. With logarithms, multiplication is reduced to simple addition, and division is reduced to simple subtrac- tion. Raising to a power is reduced to a single multiplication, and extracting a root is reduced to a single division. The common logarithm of any number is the power to which 10 must be raised in order to equal that number. |
Exponential Form | Logarithmic Form | |||||||||||||
100 | = | 10^{2} | log | 100 | = | 2.000 | |||||||||
15 | = | 10^{1.176} | log | 15 | = | 1.176 | |||||||||
10 | = | 10^{1} | log | 10 | = | 1.000 | |||||||||
7 | = | 10^{0.845} | log | 7 | = | 0.845 | |||||||||
1 | = | 10^{0} | log | 1 | = | 0.000 | |||||||||
0.1 | = | 10^{-1} | log | 0.1 | = | -1.000 | |||||||||
0.7 | = | 10^{-1.845} | log | 0.7 | = | -1.845 | |||||||||
Therefore, since | 0.015 | = | 10^{-2.176} | log | 0.015 | = | -2.176 | ||||||||
1000 | = | 10^{3} | 0.001 | = | 10^{-3} | log | 0.001 | = | -3.000 | ||||||
100 | = | 10^{2} |
it will be seen that only the direct powers of 10 have whole numbers for logarithms; also that the logarithms of all numbers lying between a power of 10, consist of a whole number and a decimal. The whole number is called the characteristic, and the decimal, the mantissa. Since the character- istic serves only to fix the location of the decimal point in the expression indicated by the log, it can be found by inspection and is not included in the log table. The following will be helpful: |
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10 | = | 10^{1} | |||||||||||||
1 | = | 10^{0} | |||||||||||||
0.1 | = | 10^{-1} | |||||||||||||
0.01 | = | 10^{-2} | |||||||||||||
0.001 | = | 10^{-3} | |||||||||||||
0.0001 | = | 10^{-4} | |||||||||||||
it is true that | |||||||||||||||
log | 1000 | = | 3 | ||||||||||||
log | 100 | = | 2 |
1. 2. 3. |
The characteristic of ant number greater than 1 is always positive and is equal to one less than the number of digits to the left of the decimal. The characteristic of any number less than 1 is always negative and is equal to one plus the numbr of zeros to the decimal. The characteristic of any number may be determined by expressing the number as a power of 10 and using this power as the characteristic of the logarithm for that number. |
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log | 10 | = | 1 | ||||||||||||
log | 1 | = | 0 | ||||||||||||
log | 0.1 | = | -1 | ||||||||||||
log | 0.01 | = | -2 | ||||||||||||
log | 0.001 | = | -3 | ||||||||||||
log | 0.0001 | = | -4 | ||||||||||||
The common sustem of logarithms has for its base the number 10, and is written log_{10} or more commonly log, since the base 10 is always implied unless some other base is specifically indicated. There are formulas however which use the natural system of logarithms. This system has for its base the number 2.718... which is represented by the Greek letter e and is always written log e. |
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A table of natural logarithms has not been included in this handbook however, since the common log of a number is ap- proximately equal to 0.4343 times the natu- ral log of the same number Conversely, the natural log of a number is approximately equal to 2.3026 times the common log of the same number. In observing the following exponential and logarithmic relationships, |
Since only the characteristic of a logarithm is ever negative, the mantissa always being a positive number, it is customary to write a con- taining a negative characteristic log as follows: log 0.7 = 1.845, or, by adding +10 to the characteristic and, in order to maintain equality, -10 at the right of the characteristic, log 0.7 = 9.845 - 10 |
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