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Question about Log Laws

  1. Apr 8, 2008 #1
    I don't know how I managed to forget this one, but I did somehow...

    If there's something like:

    e^lnx, why is that equal to just x?

    and same goes for sokmething like:

    8^log8x which is just equal to x.

    I'm just wondering how, algebraically, one could show this to be true.
  2. jcsd
  3. Apr 8, 2008 #2


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    You generally DEFINE ln(x) to be the inverse function of e^x. Or vice versa depending on which you define first. So you don't show it algebraically, it largely a matter of definition.
  4. Apr 8, 2008 #3
    well, by definition of log we have

    [tex]log_a(x)=b<=> a^b=x[/tex]

    Now lets substitute [tex] b=log_a(x)[/tex] in
    [tex]a^b=x[/tex] So:


    Or, since [tex]f(x)=a^x[/tex] and [tex]g(x)=log_ax[/tex] are inverse functions, so it means that they cancel each other out. That is

    [tex]fg(x)=f(g(x))=x=>a^{log_ax}=x[/tex] and also

    [tex] g(f(x))=log_a(a^x)=x[/tex]

    Edit: Dick was faster!
  5. Apr 8, 2008 #4


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    You are slow because you write more. Doesn't mean you think slower. I appreciate the TeX though.
    Last edited: Apr 8, 2008
  6. Apr 8, 2008 #5
    Oh I see now! Thanks so much for your help Dick and sutupidmath!
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