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Math Help - Is there a proof why this law for logarithm is such?

  1. #1
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    Is there a proof why this law for logarithm is such?

    I'd like to know whether there is a proof for the b/m laws. Not asking cos I've been asked to proved it.

    10^log x = x
    e^ln x = x

    Thanks!
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  2. #2
    MHF Contributor Unknown008's Avatar
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    What about putting log to base 10 and log to base e in front of both sides?
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    I did have a go at that, but it still doesn't make sense. I've looked through some web pages and all the books I have. They just state that it's a rule of logarithms.

    One did give the proof in the following way:

    Proof for x = 10^log x

    Let x = 10^k (Eqn. 1)
    Apply log to both sides:
    log x = log 10 ^k
    log x = k log 10
    Since log 10 = 1, therefore k = log x

    By substituting k= log x in Eqn. 1, we get x = 10^log x


    The proof's quite clear, but what if I have been asked to evaluate 10^log 3? Do I just use the law without having to prove anything? Would I still get the same marks in a test compared to someone who can explicitly prove it?
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  4. #4
    MHF Contributor Unknown008's Avatar
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    You can do it like this:

    Let:

    10^{\log\ 3} = x

    \log\ 10^{log\ 3} = \log\ x

    \log\ 3 \times \log\ 10 = \log x

    \log\ 3 = \log\ x

    Which implies that x = 3
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  5. #5
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    Hmm that seems logical. Thanks for presenting a new perspective, then.

    If any one else knows any other way(s) of solving this, I'd gladly welcome them to post more replies!
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  6. #6
    MHF Contributor Unknown008's Avatar
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    Well, you can usually even say that the log and power 10 cancel, leaving 3, but that may be too direct for a question which asks you to evaluate this.

    If for example you have a long number, were such a situation is involved, then you may use it.
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  7. #7
    Senior Member MacstersUndead's Avatar
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    Quote Originally Posted by dd86 View Post
    I'd like to know whether there is a proof for the b/m laws. Not asking cos I've been asked to proved it.

    10^log x = x
    e^ln x = x

    Thanks!
    This is analogous to showing that:

    1) 10^x and log x are inverses, and
    2) e^x and ln x are inverses.
    ((Recall that if f is an inverse of g, then f(g(x)) = g(f(x)) = x))

    One way you can show that they are inverses is to show that their graphs are symmetric about the line x = y.
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  8. #8
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    One can either first define 10^x and e^x and the define log_{10}(x) and ln(x) as their inverses or vice-versa. Either way 10^{log(x)}= x, log(10^x)= x, e^{ln(x)}= x, and ln(e^x)= x follow from the definition of "inverse" function:
    g(x) is the inverse function to f(x) if and only if both f(g(x))= x and g(f(x))= x.
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