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Math log proof

  1. Apr 18, 2004 #1
    Would someone please show me why

    [tex]{ a^{log_cb} = b^{log_ca} [/tex]

    for all a, b and c.
  2. jcsd
  3. Apr 18, 2004 #2
    I don't know if this is really a proof.. but just take the log of both sides:
    [tex]{ a^{log_cb} = b^{log_ca} [/tex]
    [tex]log_c { a^{log_cb} = log_c b^{log_ca} [/tex]
    [tex]({log_cb})({log_ca}) = ({log_ca})({log_cb}) [/tex]
    using the property that:
    [tex]{log_ca^r} = r{log_ca} [/tex]
  4. Apr 18, 2004 #3

    matt grime

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    it is a proof. perhaps to make it appear more rigorous, you could write [tex]a^{log_cb}=c^{log_c(a^{log_cb})}[/tex] and similarly for the rhs, and say that the only way for c^xto be equal to c^y is if x=y.
  5. Apr 18, 2004 #4
    Got it!

    Thanks, folks.
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