Difference between revisions of "Logarithm"

 
(fixed TeX; \log should be used for logs)
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A '''logarithm''' is a shorthand way of expressing [[exponentional notation]].  
 
A '''logarithm''' is a shorthand way of expressing [[exponentional notation]].  
 
== Introductory ==
 
== Introductory ==
The general form for a logarithm can be expressed as <math>log_x y=z</math> which means <math>x^z=y</math>. We would read this as "The logarithm of y base x is z".
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The general form for a logarithm can be expressed as <math>\log_x y=z</math> which means <math>x^z=y</math>. We would read this as "The logarithm of y base x is z".
We have <math>3^3=27</math>. To express this in [[Logarithmic notation]], we would write it as <math>log_3 27=3</math>.
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We have <math>3^3=27</math>. To express this in [[Logarithmic notation]], we would write it as <math>\log_3 27=3</math>.
 
When a logarithm has no base, it is assumed to be base 10.  
 
When a logarithm has no base, it is assumed to be base 10.  
 
==Logarithmic Properties==
 
==Logarithmic Properties==
 
These hold for all logarithms.
 
These hold for all logarithms.
*<math>log_a b^n=nlog_a b</math>
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*<math>\log_a b^n=n\log_a b</math>
*<math>log_a b+ log_a c=log_a bc</math>
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*<math>log_a b+ \log_a c=\log_a bc</math>
*<math>log_a b-log_a c=log_a \frac{b}{c}</math>
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*<math>\log_a b-\log_a c=\log_a \frac{b}{c}</math>
*<math>(log_a b)(log_c d)= (log_a d)(log_c b)</math>
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*<math>(\log_a b)(\log_c d)= (\log_a d)(\log_c b)</math>
*<math>\frac{log_a b}{log_a c}=log_c b</math>
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*<math>\frac{\log_a b}{\log_a c}=\log_c b</math>
*<math>log_a^n b^n=log_a b</math>
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*<math>\log_a^n b^n=\log_a b</math>

Revision as of 20:25, 21 June 2006

A logarithm is a shorthand way of expressing exponentional notation.

Introductory

The general form for a logarithm can be expressed as $\log_x y=z$ which means $x^z=y$. We would read this as "The logarithm of y base x is z". We have $3^3=27$. To express this in Logarithmic notation, we would write it as $\log_3 27=3$. When a logarithm has no base, it is assumed to be base 10.

Logarithmic Properties

These hold for all logarithms.

  • $\log_a b^n=n\log_a b$
  • $log_a b+ \log_a c=\log_a bc$
  • $\log_a b-\log_a c=\log_a \frac{b}{c}$
  • $(\log_a b)(\log_c d)= (\log_a d)(\log_c b)$
  • $\frac{\log_a b}{\log_a c}=\log_c b$
  • $\log_a^n b^n=\log_a b$
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