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Why b^[log (base b) a ] is = a ?

  1. Oct 11, 2008 #1
    I was wondering if anyone could show me why b^[log (base b) a ] is = a ?
     
  2. jcsd
  3. Oct 11, 2008 #2

    D H

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    Staff Emeritus
    Science Advisor

    Re: logarithms

    The equation

    [tex]b^{\log_b a} = a[/tex]

    is better written as

    [tex]b^{\log_b a} \equiv a[/tex]

    In other words, the equality is true by definition.
     
  4. Oct 11, 2008 #3
    Re: logarithms

    is it possible to prove it ? like simplify it then showed that it is equal to a ?
     
  5. Oct 11, 2008 #4

    D H

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    Staff Emeritus
    Science Advisor

    Re: logarithms

    It is not possible to prove a definition, and the standard definition of [itex]\log_b x[/itex] is

    [tex]\log_b x = y \,\Leftrightarrow \, b^y = x[/tex]
     
  6. Oct 11, 2008 #5

    Mark44

    Staff: Mentor

    Re: logarithms

    You don't prove definitions. They are the foundational building blocks that provide meanings. As an analogy, a dictionary is more or less a long list of word and definition pairs.

    A simpler example than the one you posted is: 12 inches = 1 foot. This equation defines the relationship between inches and feet and is not something that is proved.
     
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