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Do you know this property of the logarithm?

  1. Apr 29, 2012 #1
    Playing around with logarithms I found an interesting property that "log b^n(a^n) = log b(a)". Then I tried to find some kind of proof that this is right and not only a coincidence. Ι made a gereral formula for any value of both n's (α and β) so that "log b^β(a^α) = x". Therefore "a^α = b^(β*x)" ; "a = b^(β*x/α)" ; "log b(a) = β*x/α" ; "x = (α/β)*log b(a)". And therefore "log b^β(a^α) = (α/β)*log b(a)".
  2. jcsd
  3. Apr 29, 2012 #2

    Stephen Tashi

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    It's unclear what that notation is supposed to mean.

    Can you write out what "log b^n(a^n)" means in words? Or perhaps master the forums LaTex: https://www.physicsforums.com/showthread.php?t=546968
  4. Apr 29, 2012 #3


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    I think you mean "[itex]log_{b^n}(a^n)= log_b(a)[/itex]". That is, that the logarithm, base [itex]b^n[/itex], of [itex]a^n[/itex] is the same as the logarithm, base b, of a. (Of course, a and b must be positive.)

    If [itex]y= log_{b^n}(a^n)[/itex] then [itex]a^n= (b^n)^y= b^{ny}= (b^y)^n[/itex]. Can you complete it now?
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