User:Milton Beychok/Sandbox: Difference between revisions

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In [[mathematics]], the '''logarithm''' of a number to a given base is the [[Power (mathematics)|power]] or [[exponent]] to which the base must be raised in order to produce the number.<ref name=mathpage>[http://www.themathpage.com/aPreCalc/logarithms.htm Logarithms]</ref>


For example, the logarithm of 1000 to the base 10 is 3, because 10 raised to the power of 3 is 1000. As another example, the logarithm of 32 to the base 2 is 5, because 2 raised to the power 5 is 32.
The logarithm of <math>x</math> to the base <math>b</math> is written <math>\log_b\, (x)</math> or, if the base is implicit, as <math>\log (x)</math>. So, for a number <math>x</math>, a base <math>b</math> and an exponent <math>y</math>:
:<math>\text{if }b^y = x,\text{ then }y = \log_b\, (x)</math>
The <math>\log_b\,(x)</math> is a unique [[real number]] when <math>x</math> and <math>b</math> are restricted to positive real numbers and is negative for <math>0 < b < 1</math>, zero for <math>b = 1</math>, and positive for <math>b > 1</math>.
==Features of the logarithm==
An important feature of logarithms is that they reduce multiplication to addition. That is, the logarithm of the product of two numbers is the sum of the logarithms of those numbers as in this identity:
:<math>\log (x y) = \log x + \log y</math>
Logarithms also reduce division to subtraction as in this identity:
:<math>\log(x / y) = \log(x) - \log(y)</math>
And they reduce exponation to multiplication as in this identity:
:<math>\log(x^y)= y \log(x)</math>
And taking roots are reduced to division:
:<math>\log(\sqrt[y]{x}) = \frac{\log(x)}{y}</math>
The inverse of the logarithm is call the '''antilogarithm'''
and it is expressed in this identity:
:<math>\text{antilog}_b (n) = b^n </math>
Although the above practical advantages are not important for<br> numerical work today, they are used in graphical analysis (see [[Bode plot]]).
===Numerical examples===
==History==


==References==
==References==


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Revision as of 21:37, 27 October 2008


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