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Math Help - Need help proving a relationship

  1. #1
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    Need help proving a relationship

    I have to prove that

    a^log n = n^log a

    an example would be
    2^log n = n^log2 = n
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  2. #2
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    Quote Originally Posted by darken4life View Post
    I have to prove that

    a^log n = n^log a

    an example would be
    2^log n = n^log2 = n
    That isn't quite a true statement. It only works for a specific base:
    2^{log_2(n)} = n^{log_2(2)} = n

    So your statement needs to read:
    Prove: a^{log_a(n)} = n^{log_a(a)}

    This isn't so much a proof as a definition: the logarithm and exponentiation operations are functional inverses.

    What you need to do is show that each of these are equivalent to the same number.

    We know that n^{log_a(a)} = n^1 = n

    So what is a^{log_a(n)}? Well, call it x.

    Then
    x = a^{log_a(n)}

    log_a(x) = log_a \left ( a^ {log_a(n)} \right )

    log_a(x) = log_a(n)

    Since log is a 1 to 1 function, we must have x = n. Thus
    a^{log_a(n)} = n = n^{log_a(a)}

    -Dan
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