Definition

LOGARITHMS

Log Rules - h1014v1_179

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LOGARITHMS Algebra Definition Any number (X) can be expressed by any other number b (except zero) raised to a power x; that is, there is always a value of x such that X = b^x. For example, if X = 8 and b = 2, x = 3. For X = 8 and b = 4, 8 = 4^x is satisfied if x = 3/2. 4 3 2 (4³) 1 2 (64) 1 2 8 or 4 3 2 (4 1 ²)³ 2³ 8 In the equation X = b^x, the exponent x is the logarithm of X to the base b. Stated in equation form, x = log_b X, which reads x is the logarithm to the base b of X. In general terms, the logarithm of a number to a base b is the power to which base b must be raised to yield the number. The rules for logs are a direct consequence of the rules for exponents, since that is what logs are. In multiplication, for example, consider the product of two numbers X and Y. Expressing each as b raised to a power and using the rules for exponents: XY = (b^x) (b^y) = b^x+y Now, equating the log_b of the first and last terms, log_b XY = log_b b^x+y. Since the exponent of the base b (x+y) is the logarithm to the base b, Log_b b^x+y = x+y. log_b XY = x+y Similarily, since X = b^x and Y = b^y, log_b X = x and log_b Y = y. Substituting these into the previous equation, log_b XY = log_b X + log_b Y Before the advent of hand-held calculators it was common to use logs for multiplication (and division) of numbers having many significant figures. First, logs for the numbers to be multiplied were obtained from tables. Then, the numbers were added, and this sum (logarithm of the product) was used to locate in the tables the number which had this log. This is the product of the two numbers. A slide rule is designed to add logarithms as numbers are multiplied. Logarithms can easily be computed with the calculator using the keys identified earlier. MA-02 Page 64 Rev. 0

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