Machinery's Handbook, 31st Edition
LOGARITHMS
37
Property
Example
1 0 =
1 0 =
log
log
b
3
b
=
8 1 =
1
log
log
b
8
log ( ) log xy =
x
y
log( ) log log y 2 2 = +
y
+ log
b
b
b
x y ( )
100 3 ( )
log log x −
y
=
=
0 3 − log
log
log
log
10
b
b
b
r
3
= x r
x
x
x
=
log
log
log
3
log
b
b
2
2
r
5 b r Common Logarithms.—There are two standard systems of logarithms: common (base 10) and natural (base e , explained below). In general, log 10 x is written simply as log x . For example, log 100 = 2 because 10 2 = 100. Examples : log 15 = 1.176, since 10 1.176 ≈ 15 5 11 11 = = log log b The log values seen here can be found by using either log tables or a scientific calculator. Because most logarithms are irrational numbers, the values given in any example and in the tables are approximations, rounded to several decimal places. When a calculator is used, the answer should be rounded to four or five decimal places. The whole number part of a logarithm is called the characteristic ; the decimal portion is called the mantissa . In the examples above, the characteristics are 1, 2, and 3, respectively, which correspond to the power of 10 of the antilog when it is written in scientific notation: The property log( ab ) = log a + log b has been applied; the “log b ” portion (characteristic) is quickly determined, as it is simply the power of 10 (bold). The “log a ” portion (mantissa) is read from the log table, which gives logs of numbers from 1 to 10 up to a certain number of decimal places. If the log of a number less than 1 is to be found, again, the antilog is rep- resented in scientific notation to get to the answer, only now the characteristic is negative, so a subtraction is involved. For example: log 0.63 = log(6.3 × 10 –1 ) = (log 6.3) + log ( –1 ) = 0.799 + ( –1 ) ≈ –0.201 log 250 = 2.397, since 10 2.397 ≈ 250 log 4000 = 3.602, since 10 3.602 ≈ 4000 log 15 = log (1.5 × 10 1 ) = log 1.5 + log 10 1 = 0.176 + 1 ≈ 1.176 log 250 = log (2.5 × 10 2 ) = log 2.5 + log 10 2 = 0.397 + 2 ≈ 2.397 log 4000 = log (4.0 × 10 3 ) = log 4.0 + log 10 3 = 0.602 + 3 ≈ 3.602 Natural Logarithms.—In certain formulas and in some branches of mathematical analy- sis, use is made of the natural logarithm. The base of this system is given as e , which is the symbol for the irrational number that is approximately equal to 2.7182818284. (Recall that an irrational number cannot be represented by a repeating or terminating decimal.) e is the base of exponential growth phenomena such as populations and compound interest, among others. Though e was first conceptualized by John Napier (who developed logarithmic cal - culation) and developed further by Jacob Bernoulli, the use of e credits eighteenth-century Swiss mathematician Leonhard Euler (pronounced “oiler”), who developed mathematical analysis, in part, through his discovery of the so-called Euler identity, e i π = –1. It is conventional to write log e x as “ln x ”; hence: ln x = y means log e x = y , that is, x = e y So, for example, ln e = 1, since e 1 = e ; ln 1 = 0, since e 0 = 1; ln e 3 = 3 ln e = 3 ⋅ 1 = 3. And, the process of finding logarithm values in base e is the same as in base 10. Example : To represent ln 0.239 as a sum of natural logs: ln(2.39 × 10 –1 ) = ln 2.39 + ln 10 –1 = ln 2.39 + (–1)ln 10.
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