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Thread: logarithmic equation

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    logarithmic equation

    Show that alogcb =blogca
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  2. #2
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    Re: logarithmic equation

    Quote Originally Posted by Trefoil2727 View Post
    Show that alogcb =blogca
    \\\text{If }a^{\log_c(b)}=b^{\log_c(a)}\\\text{then }\log_c(b)}\log_c(a)}=\log_c(a)}\log_c(b)}
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  3. #3
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    Re: logarithmic equation

    A true "direct proof" would go the other way.
    From the obviously true log_c(b)log_c(a)= log_c(a)log_c(b), by a "property of logarithms"
    log_c\left(b^{log_c(a)}\right)= log_c\left(a^{log_c(b)}\right)
    and then take c to the power of each side.

    Of course, Plato's proof is a perfectly good "synthetic proof" where you start with what you want to prove and manipulate it (here by taking the logarithm, base c, of both sides) to produce something that is "obviously true" and each step is reversible.
    Last edited by HallsofIvy; Jan 14th 2014 at 06:11 AM.
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