Machinery's Handbook, 31st Edition
36
Functions
(a)
(b)
(c)
(d)
Fig. 2. (a) f ( x ) = x , D f = (–∞,∞); (b) f ( x ) = x 2 , D f = (–∞,∞); (c) f ( x ) = x , D f = [0,∞); (d) f ( x ) = 1/ x , D f = ( , ) ( , ). − ∞ 0 0 ∞ Logarithms
Logarithms are of value in many engineering and shop calculations because they make it possible to solve cumbersome and difficult problems that otherwise would require more complex mathematical methods. Because of this, “logs” (for short) have long been used to facilitate and shorten calculations involving multiplication, division, the extraction of roots, and obtaining powers of numbers. Since the advent of hand-held calculators, how- ever, logarithms are rarely used to do these basic operations. The Guide and Logarithms in the ADDITIONAL MATERIAL on Machinery’s Handbook 31 CD in the Machinery’s Handbook 31 Digital Edition include explanations and examples of how the log tables are used for computation. Log properties and principles are still necessary in many areas. Logarithmic growth and its inverse, exponential growth (and decay), are essential to investigating processes in technical fields and science, in general. The main principles and properties of logarithms are covered here, along with representative examples. In most cases, a calculator is used to arrive at the answers. Meaning of Logarithm.—The logarithm of a given number is the exponent to which a stated base must be raised to produce the given number. A formulaic definition of loga - rithm is: log b y x y b x = = means which is read, “The logarithm of x in base b is y ; that is, b raised to the y power equals x .” y is the logarithm and x is the antilogarithm (“antilog”). Base b is always greater than 1. The antilog must be positive, since a positive number b raised to any power cannot give zero or a negative number. Some examples: log 2 8 = y means 2 y = 8, so y = 3 log b 100 = 2 means b 2 = 100, so b = 10 log 3 x = –1 means 3 –1 = x , so x = 1/3 Properties of Logarithms.—The definition along with exponent rules covered previ - ously lead to the properties of logarithms , given here with examples. When no base is shown, base 10 is implied. (See Common Logarithms on page 37.)
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